Washington (CNN) President Barack Obama said Wednesday that it was past time for the U.S. to re-establish diplomatic relations with Cuba as he announced that the two countries were reopening their embassies after more than 50 years.

"When the United States shuttered our embassy in 1961, I don't think anyone thought it would be more than half a century before it reopened," he said in remarks from the White House Rose Garden.

Earlier Wednesday in Havana, a U.S. diplomat delivered a note from Obama to Cuban President Raul Castro restoring diplomatic ties.

The short ceremony at the Cuban Foreign Ministry in Havana ended 54 years of broken relations that began during the Eisenhower administration. Jeffrey DeLaurentis, the chief of mission at the U.S. Interests Section, delivered the note.

Castro also sent a letter to Obama.

"We want to develop a friendship between our two nations that is based on the equality of rights and the people's free will," Castro said in the letter, read on state-run TV.

He went on to say that Cuba hopes to resolve differences with the United States through peaceful means, that each nation must respect the territorial integrity of the other and they should not interfere in each other's political affairs.

However, the Cuban Foreign Ministry indicated that hurdles still remained in the thaw of U.S.-Cuba relations due to the embargo that the U.S. has imposed on Cuba.

"There could be no normal relations between Cuba and the United States as long as the economic, commercial and financial blockade continues to be fully implemented, causing damage and scarcities to the Cuban people," reads a statement received by CNN. "The blockade is the main obstacle to the development of our economy; it is a violation of International Law and affects the interests of all countries, including those of the United States."

Obama has relaxed several of the prohibitions on trade and travel that have existed between the two countries, but many remain in place and can only be removed by legislation.

Obama called Wednesday for Congress to lift the embargo that prevents Cubans from traveling or doing business in Cuba.

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"Americans and Cubans alike are ready to move forward. I believe it's time for Congress to do the same," he said. "We've already seen members from both parties begin that work."

Reaction on Capitol Hill

But Congress, controlled by Republicans, has shown little sign that it intends to end the embargo. Several GOP candidates for President expressed their opposition to the shift in policy.

Texas Sen. Ted Cruz was quick to criticize Obama's move to reopen embassies.

"It's unacceptable and a slap in the face of a close ally that the United States will have an embassy in Havana before one in Jerusalem," tweeted Cruz, who is seeking the 2016 Republican presidential nomination.

In a statement, Florida Sen. Marco Rubio vowed to oppose the confirmation of an ambassador to Cuba until the Castro regime makes several concessions, including "securing greater political freedoms for the Cuban people."

And former Texas Gov. Rick Perry said the Cuban government had made no changes that suggest re-engagement would benefit Cuban citizens.

"The Cuban people today are not any freer politically or economically, and President Obama has failed to account for what the Castro regime has done in the last several years that warrants such an enormous shift in a longstanding U.S. policy," he said.

But Obama also has supporters on Capitol Hill.

Democratic Tennessee Rep. Steve Cohen quickly put out a statement saying that the U.S. and Cuba working together was the only way to address concern about Cuba's behavior.

"These decisions will also help promote human rights and freedom in a country where we have had little influence for too long," he said. "I am proud of the quick, but no doubt difficult, work our President has done to chart a new course in our relationship with Cuba, and I wholeheartedly support his efforts to tear down the walls that separate us."

In his Rose Garden speech, which was carried live on Cuba's state-run Cubavision network, Obama noted that many Cubans have called for increased U.S. engagement with their country, and said it was time to look toward the future.

"You can't hold the future of Cuba hostage to what happened in the past," Obama said, quoting a Cuban resident.

The President agreed, pointing out that the Eisenhower administration had severed ties with Cuba in the same year that Obama had been born.

"You don't have to be imprisoned by the past," he said. "If something isn't working, we can and will change."

But after his announcement, protesters in Miami spoke out against the move, indicating at least some pockets of the Cuban community are not supportive of his policy.

Obama also said that working more closely with Cuba will allow the U.S. to address human rights issues that go against America's policies.

"We will not hesitate to speak out when we see actions that contradict those values," Obama said. Critics, however, have expressed doubt on that score.

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The next steps

Diplomatic relations will be officially re-established on July 20, according to a statement from the Cuban Foreign Ministry. The country will also open its embassy in Washington that day.

Secretary of State John Kerry announced from Vienna Wednesday that he will travel to Cuba this summer to take part in the formal reopening of the U.S. Embassy. It will be the first visit by a U.S. secretary of state since 1945.

He called the move "an important step on the road to restoring fully normal relations between the United States and Cuba," adding that "coming a quarter of a century after the Cold War, it recognizes the reality of the changed circumstances and it will serve to meet a number of practical needs."

Kerry acknowledged that the United States and Cuba continue to have "sharp differences over democracy, human rights and related issues," but added that the two countries have identified areas for cooperation that will include law enforcement, transportation, emergency response, environmental protection, telecommunications and migration.

The opening of embassies is the culmination of Obama's initiative to thaw relations begun in December. Travel restrictions have been loosened since that time and some new economics ties have been established. The U.S. removed Cuba from its state sponsors of terror list in May.