The family of Otto Warmbier, the American student who died in 2017 after being imprisoned in North Korea, spoke out on Friday about President Trump’s refusal to blame Kim Jong-un for the death of their son, saying the North Korean leader was responsible.

The statement was the first reaction from Mr. Warmbier’s parents, Fred and Cindy Warmbier, to the president’s comments during a summit meeting with Mr. Kim in Vietnam this week that he would take him “at his word” that he did not know about Mr. Warmbier’s treatment while in prison.

“We have been respectful during this summit process,” the Warmbiers said. “Now we must speak out.”

“Kim and his evil regime are responsible for the death of our son Otto,” they said. “Kim and his evil regime are responsible for unimaginable cruelty and inhumanity. No excuses or lavish praise can change that.”

Mr. Trump’s remarks had set off anger and sympathy for the young man’s family among political leaders in the United States on Thursday, as he abruptly wrapped up his meeting with Mr. Kim.

Mr. Trump said that he and Mr. Kim had discussed Mr. Warmbier and that Mr. Kim felt “badly” about what happened to the American. Mr. Warmbier died shortly after he was released to the United States, with doctors saying he had sustained a catastrophic brain injury.

“He tells me that he didn’t know about it, and I will take him at his word,” Mr. Trump said.

“I don’t believe that he would have allowed that to happen, it just wasn’t to his advantage to allow that to happen,” he added. “Those prisons are rough, they’re rough places, and bad things happened. But I really don’t believe that he, I don’t believe that he knew about it.”

Mr. Trump’s comments, which echoed other remarks he has made defending assertions by autocratic foreign leaders that conflict with the assessments of his own intelligence agencies, prompted anger from Democrats and cautiously critical responses from several Republicans.

The Trump administration has stood by Saudi Arabia’s day-to-day leader, Crown Prince Mohammed bin Salman, even after American intelligence agencies concluded that the prince most likely ordered the killing of the journalist Jamal Khashoggi, a United States resident, in the Saudi consulate in Istanbul in October.

In a remarkable statement in November, Mr. Trump said, “It could very well be that the crown prince had knowledge of this tragic event — maybe he did and maybe he didn’t!”

“We may never know all of the facts surrounding the murder of Mr. Jamal Khashoggi,” the president added. “In any case, our relationship is with the Kingdom of Saudi Arabia.”

On Thursday, Representative Nancy Pelosi, the Democratic speaker of the House, recalled Mr. Trump’s previous remarks defending President Vladimir V. Putin of Russia and challenging the conclusion of his own intelligence agencies that Moscow interfered in the 2016 presidential election.

“It’s strange,” she said at her weekly briefing. “I don’t know. There is something wrong with Putin, Kim Jong-un — in my view, thugs — that the president chooses to believe.”

Mr. Trump and Mr. Kim ended their second summit meeting on Thursday after failing to agree on steps toward nuclear disarmament or measures to ease tensions on the Korean Peninsula.

At the Capitol, Senator Mitch McConnell of Kentucky, the majority leader, praised Mr. Trump’s “smart” handling of North Korea and said he “should be commended for walking away.” He did not comment on Mr. Trump’s remarks about Mr. Warmbier, CNN reported.