Havana (CNN) U.S. President Barack Obama put the authoritarian government in Havana on the spot Monday, taking questions from reporters and insisting that his Cuban counterpart also deliver answers to pointed queries on human rights, political prisoners and economic reforms.

Though they both acknowledged deep disagreements on these issues, the two leaders found common ground on the topic of the economic embargo on Cuba, which both want lifted. Obama went so far as to declare that "the embargo's going to end," though he couldn't say when.

In an extraordinary sign of the shifting attitudes, Castro was willing to answer one question on why his regime was keeping Cubans incarcerated for expressing anti-government views. But his response only underscored the schisms between himself and Obama.

"Did you ask if we had political prisoners? Give me a list of political prisoners and I will release them immediately," Castro said defensively when asked by CNN's Jim Acosta why his government was incarcerating dissidents.

Unaccustomed to press conferences, Castro at first appeared confused at whether the question was directed to him and later asked for it to be repeated as he juggled the headphones he wore to hear its translation.

Later, Castro delivered a litany of areas where he said the U.S. was failing, from inadequate health care to lower pay for women. He ended the unprecedented question-and-answer session after a second inquiry on human rights, saying he'd said "enough."

Obama, meanwhile, appeared to relish putting Castro on the spot, winking at the assembled journalists when it appeared Castro wasn't going to answer his question.

"Excuse me?" Obama said to prompt the Cuban leader.

In his own message on human rights, Obama defended his decision to come to Cuba even as government dissent is punished.

"We have decades of profound differences," Obama said when asked what his message on human rights was during his "frank conversation" on the issue with Castro. "I told President Castro that we are moving forward and not looking backwards."

"We will continue to stand up for basic principles that we believe in," said Obama, who at points insisted that Castro answer the questions posed to him by American journalists. "America believes in democracy. We believe that freedom of speech, freedom of assembly, are not just American values but universal values."

Obama was speaking following an hours-long meeting with Castro at the Palace of the Revolution in Old Havana, his third meeting with the Cuban leader since work began to reopen diplomatic ties to the island.

Castro, making a statement before the reporters' questions, said that work toward improving economic conditions in his country was progressing. But he added that a longstanding trade embargo prevents a full restoration of ties.

"Much more could be done if the U.S. blockade could be lifted," Castro said. "The most recent measures adopted by his administration are positive but insufficient."

Obama agreed that the restrictions would eventually be removed, adding that, "The path that we're on will continue beyond my administration. The reason is logic."

Photos: President Obama visits Cuba U.S. President Barack Obama attends a baseball game in Havana, Cuba, with his family and Cuban President Raul Castro, right, on Tuesday, March 22. The Cuban national team was playing an exhibition against Major League Baseball's Tampa Bay Rays. Obama is the first U.S. President to visit Cuba since Calvin Coolidge in 1928. Hide Caption 1 of 19 Photos: President Obama visits Cuba Tourists in a Havana antique shop watch Obama give a speech on March 22. Hide Caption 2 of 19 Photos: President Obama visits Cuba Obama waves to the crowd before delivering his speech at the Grand Theater in Havana on March 22. In his speech, Obama urged Cubans to look to the future with hope, casting his historic visit as a moment to "bury the last remnants of the Cold War in the Americas." Hide Caption 3 of 19 Photos: President Obama visits Cuba First lady Michelle Obama and her daughter Malia stand with children of U.S. Embassy workers after dedicating a bench and two magnolia trees at a small park in Havana on March 22. Hide Caption 4 of 19 Photos: President Obama visits Cuba The Obamas pose with Castro before a state dinner in Havana on Monday, March 21. Hide Caption 5 of 19 Photos: President Obama visits Cuba U.S. Secretary of State John Kerry, right, speaks with Cuban First Vice President Miguel Diaz-Canel ahead of the state dinner on March 21. Hide Caption 6 of 19 Photos: President Obama visits Cuba Castro tries to lift up Obama's arm at the end of a joint news conference on March 21. Hide Caption 7 of 19 Photos: President Obama visits Cuba Obama meets with Castro in Havana on March 21. Hide Caption 8 of 19 Photos: President Obama visits Cuba Michelle Obama speaks with Cuban girls during a Let Girls Learn roundtable in Havana on March 21. Hide Caption 9 of 19 Photos: President Obama visits Cuba Obama lays a wreath at the Jose Marti monument in Havana's Revolution Square on March 21. "It is a great honor to pay tribute to Jose Marti, who gave his life for independence of his homeland," Obama wrote after he laid the wreath. "His passion for liberty, freedom, and self-determination lives on in the Cuban people today." Hide Caption 10 of 19 Photos: President Obama visits Cuba Backdropped by a monument depicting revolutionary hero Che Guevara, Obama listens to the U.S. national anthem during the wreath-laying ceremony on March 21. Hide Caption 11 of 19 Photos: President Obama visits Cuba Obama and Castro review troops before bilateral meetings on March 21. Hide Caption 12 of 19 Photos: President Obama visits Cuba Obama walks up the stairs of the Palace of the Revolution on March 21. Hide Caption 13 of 19 Photos: President Obama visits Cuba Obama and members of the first family take a walking tour of a Havana cathedral on Sunday, March 20. Hide Caption 14 of 19 Photos: President Obama visits Cuba The first family stops to look at a painting of Abraham Lincoln in the Museum of the City of Havana on March 20. Hide Caption 15 of 19 Photos: President Obama visits Cuba Obama's convoy drives along the Malecon sea wall on March 20. Hide Caption 16 of 19 Photos: President Obama visits Cuba The President and first lady greet families of embassy personnel on March 20. Hide Caption 17 of 19 Photos: President Obama visits Cuba Obama waves shortly after arriving at Jose Marti International Airport. Hide Caption 18 of 19 Photos: President Obama visits Cuba Obama and his family exit Air Force One on March 20. Hide Caption 19 of 19

Obama started his morning laying a wreath by the Jose Marti memorial, a massive monument to the Cuban revolutionary leader where a Cuban military band played the "Star Spangled Banner," another in a series of previously unthinkable moments that marks this week's visit.

The sight of a sitting American president setting foot on the island was a novelty for most Cubans. The last U.S. leader to visit was Calvin Coolidge, who voyaged into Havana Harbor on a battleship in 1928.

In an elaborate welcome ceremony, Obama and Castro met with smiles and brief conversation before moving down long hallways lined with Cuban troops. Obama was overheard telling the Cuban leader he enjoyed his tour of Havana Sunday night along with his family.

He also told Castro he had a "great" dinner at a "paladar" -- one of hundreds of privately run restaurants that only recently became permissible in the state-run economy. Those types of businesses, along with new investments from American firms, give U.S. officials hope that Cuba is on a path to open its economy after decades of isolation.

The meeting provides Obama and his aides another reality check on their mission to extract reforms from Castro. Until this point, there have been few signs that the government here is willing to work as quickly as the Obama administration hoped in opening the state-run economy and improving human rights.

Photos: Illustrious visitors to Cuba Former U.S. President Jimmy Carter, left, visits Cuba in 2002. He was invited by Cuban President Fidel Castro, right, and the two men shook hands at the State Council in Havana. This month, President Barack Obama will be the first sitting U.S. leader to visit Cuba since Calvin Coolidge in 1928. But the nation has been far from isolated. Here are other world leaders, religious figures and big-time celebrities who have been to the Caribbean island. Hide Caption 1 of 15 Photos: Illustrious visitors to Cuba Shortly after being released from prison in 1990, Nelson Mandela visited Cuba and shook hands with Castro. Castro and Mandela were friends. In 1994, when Mandela became the first black President of South Africa, Castro was a guest of honor at his inauguration. Hide Caption 2 of 15 Photos: Illustrious visitors to Cuba In 1989, just two years before the Soviet Union collapsed, Castro met with Soviet leader Mikhail Gorbachev. Cuba and Russia enjoy a long friendship going back to the Cold War. Hide Caption 3 of 15 Photos: Illustrious visitors to Cuba After the collapse of the Soviet Union, no Russian leader visited Cuba until 2000, when President Vladimir Putin traveled to the island to meet with Castro. Putin returned in 2014, meeting with Castro and his brother Raul. "Cooperation with the Latin American nations is one of the key orientations and prospects of Russian foreign policy," Putin said afterward. Hide Caption 4 of 15 Photos: Illustrious visitors to Cuba Former Venezuelan President Hugo Chavez was one of Fidel Castro's closest allies. Their friendship spanned almost two decades. Chavez visited Cuba for the first time in 1994, shortly after his release from jail for his involvement in a failed government coup. Chavez visited Cuba many times in the years after that visit. He also received medical treatment on the island until his death in 2013. Hide Caption 5 of 15 Photos: Illustrious visitors to Cuba Cuba has hosted multiple visits by Iranian heads of state. Here, Cuban President Raul Castro, left, attends a welcoming ceremony with Iranian President Mahmoud Ahmadinejad in 2012. During his trip, Ahmadinejad called on developing countries to unite against "imperialism and capitalism." Hide Caption 6 of 15 Photos: Illustrious visitors to Cuba Fidel Castro enjoyed a close relationship with many Middle Eastern heads of state. One of them was Iraqi President Saddam Hussein. Hussein's first visit to Cuba was in 1979, when he was vice president. Hide Caption 7 of 15 Photos: Illustrious visitors to Cuba Cuba has also welcomed three Popes since the revolution. Pope John Paul II paid the first-ever papal visit to Cuba in 1998. He was greeted personally by Fidel Castro. The pontiff toured the island nation for five days. Hide Caption 8 of 15 Photos: Illustrious visitors to Cuba Pope Benedict XVI visited Cuba 14 years after his predecessor. The trip came a little less than a year before his retirement. During his visit, the Pope told the audience that he sought to emphasize "the importance of faith," highlighting the need for good relations between the church and the island nation. Hide Caption 9 of 15 Photos: Illustrious visitors to Cuba No Pope has been to the island more times than Pope Francis. Even though he has been the leader of the Catholic Church for less than three years, Pope Francis has already visited Cuba twice. His first visit was in 2015. Francis called on the communist nation to "open itself to the world," while praising its recent restoration of diplomatic ties with the United States. In 2016, Cuba served as a backdrop to the first meeting between the heads of the Catholic and Russian Orthodox churches since the Great Schism in 1054. Hide Caption 10 of 15 Photos: Illustrious visitors to Cuba In recent years, Cuba has seen a star-studded cast of visitors. Rapper Jay Z and his wife, pop star Beyonce, caused a stir during their trip to the island in 2013. Their visit was so heavily criticized that the Treasury Department's Office of Inspector General had to get involved. The Treasury Department deemed their trip did not violate any U.S. sanctions laws that were in place during the visit. Hide Caption 11 of 15 Photos: Illustrious visitors to Cuba American movie director Steven Spielberg spent four days in Cuba in 2002. The trip, which had been authorized by the U.S. government as a cultural exchange, served as a way for the filmmaker to showcase some of his movies and meet with some Cuban filmmakers. Spielberg also dined with Fidel Castro, discussing arts, politics and history. Hide Caption 12 of 15 Photos: Illustrious visitors to Cuba In the early 2000s, when former soccer star Diego Maradona was battling his cocaine addiction, the Argentine sought treatment in Cuba. He and Fidel Castro have been close ever since. Maradona has visited the island and met with Castro multiple times. The two have even exchanged letters. In 2015, it was a letter to Maradona that quelled rumors the Cuban leader had passed away. Hide Caption 13 of 15 Photos: Illustrious visitors to Cuba In 2001, American actor Kevin Costner went to Cuba to give Fidel Castro a private screening of Costner's movie "Thirteen Days." The movie dealt with the subject of the Cuban Missile Crisis. Hide Caption 14 of 15 Photos: Illustrious visitors to Cuba "How could I miss this opportunity? Fidel, much more than a man, is a great idea." Those were French actor Gerard Depardieu's first words upon arriving in Havana. The actor has been to the island multiple times, including a celebration for Castro's 80th birthday in 2006. Hide Caption 15 of 15

The vast differences between the Obama administration and the Castro regime were no less evident Sunday as Havana hurried to prepare for the U.S. president's arrival. As Obama was en route, scores of anti-Castro dissidents from the group Ladies in White were arrested and detained after their weekly protest in Havana. CNN witnessed dozens of protesters being driven from the site in buses.

The group consists largely of women who have been arrested and imprisoned for speaking out against the government here.

Jose Daniel Ferrer, a Cuban dissident who was imprisoned for eight years beginning in 2008, said Obama could harness the attention from his trip to make a loud demand for political reform, citing President Ronald Reagan's demand in 1987 for the Soviet Union to "tear down this wall."

But Ferrer, speaking with CNN in his home Sunday, conceded that even incremental change is beneficial to the island's politically oppressed citizenry.

"It's a support gesture, it's a solidarity gesture for the struggle that we are taking forward," he said. "In Cuba we have to tear down many walls so that the Cuban people can live with dignity, with rights, and so that they can prosper."

Ferrer is among the dissidents meeting Obama on Tuesday.

JUST WATCHED Here's what Cubans think about Obama's visit Replay More Videos ... MUST WATCH Here's what Cubans think about Obama's visit 01:39

Just before Obama stepped from Air Force One Sunday -- carrying an umbrella as a persistent rain fell on the tarmac -- he sent a message to Cubans on a platform that until recently would have been unheard of in the repressive regime.

"¿Que bolá Cuba?" he wrote on Twitter, using an informal Cuban greeting. "Just touched down here, looking forward to meeting and hearing directly from the Cuban people."

At the airport, Obama was met by a low-wattage greeting party that didn't include Castro, a fact GOP presidential front-runner Donald Trump and other Republicans deemed a snub.

"I've flown on that wonderful airplane that landed there," said former Sen. Mel Martinez, a Republican from Florida. "I always dream of that plane returning to Cuba, but I always thought that it would happen under different conditions, when there was a Cuban president who would meet him at the airport, and show him that kind of respect, but also that was elected by the Cuban people."

White House officials, however, said they felt no umbrage and noted that heads of state don't often come to the airport to greet visiting leaders.

Obama also faced criticism from Cuban-American Ted Cruz, a Texas senator and another Republican presidential candidate who said he wouldn't visit under the current political conditions.

In a Politico op-ed, Cruz called the visit "so sad, and so injurious to our future as well as Cuba's, that Obama has chosen to legitimize the corrupt and oppressive Castro regime with his presence on the island."

Cruz wrote that Obama's visit sends a message to political prisoners in Cuba that "the world has forgotten about you."

JUST WATCHED President Obama: This is a historic visit, opportunity Replay More Videos ... MUST WATCH President Obama: This is a historic visit, opportunity 00:52

"There will be no mojitos at the U.S. Embassy for them," the Texas senator wrote, jabbing Obama's visit, which included a formal welcoming ceremony and a state dinner at the Revolutionary Palace.

On Sunday, Obama's interactions with everyday Cubans were carefully calibrated. He toured the Cathedral of the Virgin Mary of the Immaculate Conception, greeting Cardinal Jaime Ortega, a key proponent of improving ties between the U.S. and Cuba. Crowds had gathered in the faded colonial streets of Old Havana to glimpse Obama and his family as they passed through on foot.

The presidential trip to Havana is the culmination of a three-year effort to restore ties to the island, which sits 90 miles from Key West, Florida, but has long been off-limits for most American visitors. For decades, the island was regarded as a Cold War adversary, a forbidden place run by bearded strongmen that residents fled on makeshift rafts.

With his family in tow, Obama hopes to change that perception, highlighting the country's emerging private economy and meeting with outspoken opponents of the Castro regime. Before he departed, Obama met with Cuban-Americans in a bid to garner support for his diplomatic thaw, which is still met with skepticism among many in the large Cuban diaspora in South Florida and elsewhere. Many Republicans, particularly GOP presidential candidates, have also lambasted the move.

Photos: Scenes from Cuba A 1955 Chevy Bel Air is one of thousands of old American cars that still fill the streets of Havana. Cubans lucky enough to keep the cars running now ferry tourists around town for about $40 an hour -- twice what the average Cuban earns in a month. Hide Caption 1 of 10 Photos: Scenes from Cuba The sun sets on the rooftops of the Old City. Eighty percent of the buildings in Havana were constructed between 1900 and 1958, before the American embargo took effect. Many are now in urgent need of repair. Hide Caption 2 of 10 Photos: Scenes from Cuba A city of over 2 million people, Havana may be the world's sexiest ruin. Many Cubans are worried about the social and environmental effects of an influx of tourists. Hide Caption 3 of 10 Photos: Scenes from Cuba A band and a group of stilt dancers whip around the streets of Old Havana, attracting crowds of visitors. Though Americans are finally normalizing relations with Cuba, tourists from South America, Canada and Europe have been visiting for generations. Hide Caption 4 of 10 Photos: Scenes from Cuba The streets of Old Havana are full of texture and color, and Cubans are fiercely proud of their island's soul. "Freedom, for me, goes beyond material things," said one translator. Hide Caption 5 of 10 Photos: Scenes from Cuba Local teens play soccer in the city after school. Cubans enjoy free education, as well as free health care. With a vast network of family doctors, they have lower infant mortality than Americans, and, according to some statistics, longer lifespans. Hide Caption 6 of 10 Photos: Scenes from Cuba It's common to find abandoned construction sites around Havana, some overgrown with vegetation, giving each site a form and character of its own. Hide Caption 7 of 10 Photos: Scenes from Cuba Viñales, Cuba, is a UNESCO World Heritage Site and home to some of the world's most fertile soil, perfect for producing the tobacco used to make the country's prized cigars. Hide Caption 8 of 10 Photos: Scenes from Cuba A farmer drives his oxen through the small-town streets of Viñales. Visitors to the countryside see another side of Cuba. Hide Caption 9 of 10 Photos: Scenes from Cuba The southernmost point in the United States, Key West sits just 90 miles from the Cuban shoreline. Cuba gets 3 million tourists a year; the state of Florida receives 92 million. Hide Caption 10 of 10

With relaxed restrictions on who can travel to Cuba, many more Americans will now be able to follow the Obamas' lead. And administration officials hope improved economic ties can foster a new dynamic between individual Cubans and Americans.

While Monday is focused largely on the Cuban government — including a state dinner at the Revolutionary Palace Monday night — Obama during his trip will also peel away for less formal encounters, allowing the presidential spotlight to also shine on ordinary Cubans living in a new era, including during an address to the Cuban people broadcast on state television and the meeting with anti-Castro dissidents.

Before he departs Tuesday, Obama will watch the Cuban national baseball team play the Tampa Bay Rays, in town for an exhibition game as U.S. Major League Baseball works to update immigration rules for Cuban players.