Puerto Rico Gov. Ricardo Rossello refused to relinquish power Wednesday despite rising calls for the young, telegenic head of state’s resignation, a crisis fueled by a digital-age chat scandal that has thrown the U.S. territory into political chaos.

With protesters gathered in the streets of San Juan for the 12th day, Mr. Rossello’s office said that he had not resigned and that he remains in Puerto Rico, countering speculation that he had left the island.

“As he stated yesterday, he is in a process of reflection and of listening to the people,” said the statement by Secretary of Public Affairs Anthony Maceira.

Meanwhile, the Puerto Rico House of Representatives moved closer to impeachment, announcing Wednesday that a panel of three lawyers recommended a trial after determining that five offenses were committed in the notorious chat on the Telegram app, according to Telemundo and El Nuevo Dia.

Mr. Rossello’s chief of staff, Ricardo Llerandi Cruz, submitted his resignation effective July 31, citing the threats to his family, as reported by CNN.

The 889-page chat between Mr. Rossello and 11 top aides and cabinet members, obtained by Puerto Rico’s Center for Investigate Journalism, showed government officials doing partisan political work on government time and making vulgar comments about female politicians and singer Ricky Martin, who is gay.

Carlos A. Suarez Carrasquillo, lecturer at the University of Florida’s political science department, said the chat mocked virtually every sector of Puerto Rican society, including victims of Hurricane Irma and Hurricane Maria.

“They even made fun of members of his own party, the LGBT community, the poor, you name it,” Mr. Suarez said. “There was no segment of the Puerto Rican population that was free from some sort of mockery in these chats.”

The transcript ignited mass protests among Puerto Ricans already dissatisfied with the slow pace of hurricane recovery; austerity measures introduced to grapple with the $73 billion debt, and the FBI arrests earlier this month of two former administration on corruption charges.

Even Mr. Martin, a folk hero on the island, joined the protests, which swelled to an estimated 500,000 last weekend. On social media, critics referred to the scandal as “chatgate” and “RickyLeaks.”

“This is the cumulative effect of many years,” said Mr. Suarez. “This moment now where the island finds itself is because these chats justified what many have suspected, that the political class on the island operated in a ruthless fashion and was potentially corrupt. So all the things that people imagined—actually for the first time, there’s evidence that points to it.”

In a televised statement, Mr. Rossello apologized for his comments, which included calling former New York City Council Speaker Melissa Mark-Viverito a “whore,” and said he would not seek reelection, even as the crowds outside the governor’s residence demand his “renuncia,” or resignation.

Both Democrats and Republicans have denounced him. On Monday, President Trump called Mr. Rossello a “terrible governor,” while New York City Mayor Bill de Blasio and Attorney General Letitia James said the governor should resign.

“While Puerto Ricans attempt to heal from Hurricane María, he mocks victims for a cheap laugh,” said Mr. de Blasio, a 2020 Democratic presidential contender. “Governor Rossello should resign immediately.”

The 40-year-old son of former Puerto Rico Gov. Pedro Rossello, who holds a B.A. from MIT and a Ph.D. in biomedical engineering from the University of Michigan, was elected to the state’s highest office in 2016 on the New Progressive Party ticket in his first bid for public office.

Mr. Rossello argued in a Monday interview that it would be wrong to run from his responsibilities even as he struggled to come up with the name of anyone who still supports him.

“There are other folks that have established people in the legislature and people in the Senate, as well, have supported the state,” Mr. Rossello told Fox News. “They have supported the fact that I’m not going to run and that I shouldn’t seek reelection, but they have established it’s important to follow the rule of law.”

No Puerto Rico governor has resigned from office before, but no head of state has won election to a second term since Mr. Rossello’s father did so in 1996, an indicator of the U.S. territory’s mounting political and economic woes.

About 14 percent of the population left the island from 2010-18, driven by a 12-year recession and Hurricane Maria, the 2017 Category 4 storm that wiped out the power grid and caused $102 billion in damage.

The person expected to succeed Mr. Rossello is a member of his own administration, Secretary of Justice Wanda Vazquez.

A resignation may not signal the end of the demonstrations. Protesters, who include members of the pro-statehood PNP, have announced plans to descend Thursday on the financial district whether the governor leaves office or not.

“What some of these organizers have voiced is that having the governor step down is not enough,” Mr. Suarez said. “They have argued that it’s a larger issue, in terms of his political party, and some have argued it’s an issue of the political class.”

The next target may be members of the Financial Oversight and Management Board created by Congress in 2016, the drive behind some of the austerity measures.

“Some have said that if removing the governor was possible, removing members of the advisory oversight board will be even easier,” Mr. Suarez said. “What will be interesting is whether the movement will have the staying power to achieve this, or if any fractures will eventually show up. So we’ll see.”

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