Trump's tweets come as the unrest on the island grew more violent Wednesday night, when police fired tear gas on those marching in San Juan.
The president tweeted the people of Puerto Rico "are great" but "much of their leadership is corrupt" -- a charge he's made before.
He noted Rosselló was "under siege" and, as he has before, lambasted San Juan's mayor, Carmen Yulín Cruz, saying she "is a despicable and incompetent person who I wouldn’t trust under any circumstance."
The mayor has made clear in recent days that she's on the side of the protesters, and she was also among the targets of the profanity-filled secret Telegram chat group that included the governor and other top officials that helped spark the protests.
Cruz responded to Trump in a Facebook post on Thursday afternoon.
“President Trump you never got it; and you never will. This is not about you; this is about the dignity of the Puerto Rican people...what is happening in Puerto Rico is that a people united by a profound sense of dignity are on the streets protesting corruption and a misogynist, homophobic, two-faced governor," Cruz wrote.
In a new statement issued Thursday morning, Rossello acknowledged the widespread protests, saying they "have not gone unnoticed for me, my family and certainly for anyone in Puerto Rico."
He said he respected the demonstrations "not only as a democratic exercise, but as a natural manifestation of the anger at recent events," but criticized what he said were the minority of protesters who "chose the wrong methods and violence, even with the use of weapons, Molotov cocktails and other explosives, causing injuries and impacting officers of the Puerto Rico Police."
"This challenge to law and order will be addressed accordingly," he said of the violent demonstrators.
"In the past few days I have apologized to the Puerto Rican people and that request remains alive. I have the commitment, stronger than ever, to carry out the public policy for which we have worked so hard in all areas of government. I recognize the challenge before me for recent controversies, but I firmly believe that it is possible to restore confidence and that we can, after this painful and painful process, achieve reconciliation," he said.
Julio Ricardo Varela, co-host of the political podcast, "In The Thick" and founder of Latino Rebels, called Rosselló "tone deaf" in an interview on ABC News Live's The Briefing Room.
"There’s a bigger deeper thing here that’s happening in Puerto Rico that goes beyond group texts. It’s about decades of corruption, decades of a political class that’s mostly white, and male, and privileged, sort of ignoring the real needs of the island," Varela said. "This is colonialism in action. What we’re seeing right now is the Puerto Rican people finally trying to figure out what the next step is."
“Rosselló is totally tone deaf right now. He’s not listening to the political will at all and he has no allies. He has no allies anywhere," Varela added.
The uproar began Saturday, after the nonprofit journalism group Center of Investigative Journalism published nearly 900 pages of conversations from a leaked group chat between Rosello and several top aides that detail efforts to manipulate public narratives, operations to discredit negative press coverage and criticism of opposition leaders.
The conversations, made through the Telegram app, also contain sexist, homophobic and misogynistic comments from the members of the group, according to the report. These messages have not been independently authenticated by ABC News. After the revelation of the messages, Rossello announced the resignation of a number of government officials including Luis Rivera Marin, the secretary of state.
ABC News' Victoria Moll Ramirez contributed to this report.