• President Trump and Kim Jong-un, the leader of North Korea, failed to reach a deal on denuclearization during their second summit meeting in eight months. “Sometimes you have to walk,” Mr. Trump said after the talks broke down.

• Mr. Trump said Mr. Kim was willing to close some but not all nuclear sites in North Korea in exchange for the lifting of all international sanctions.

• The leaders were scheduled to have lunch together and sign a joint agreement, but those plans were abruptly called off. After a news conference, Mr. Trump boarded Air Force One to return to Washington.

Lifting punishing international sanctions that limit North Korea’s ability to import oil, and to export lucrative goods including coal and seafood, is the North’s primary goal in any negotiation. As a result, the United States sees the sanctions as a critical bargaining chip.

Mr. Trump said he and Mr. Kim discussed the closure of North Korea’s main nuclear complex in Yongbyon, and Mr. Kim expressed a willingness to allow the facility to be dismantled.

“He would do that but he wants the sanctions for that,” Mr. Trump said. “As you know, there’s plenty left after that. I just felt it wasn’t good.”

Yongbyon is the North’s largest facility, but not its only one. At his news conference, Mr. Trump acknowledged that the country had another uranium enrichment plant. North Korea has long been suspected of having uranium enrichment capabilities beyond Yongbyon.

Prime Minister Shinzo Abe of Japan fully backed President Trump’s decision to walk from his summit meeting with Kim Jong-un, North Korea’s leader, without an agreement, while South Korea called the move regrettable.

“With the strong determination to realize the denuclearization of the Korean Peninsula, President Trump did not make an easy compromise,” Mr. Abe said, speaking to reporters after a phone call with the American president. “I fully support Mr. Trump’s decision.”

Mr. Abe reiterated his desire to meet the North Korean leader and discuss the issue of Japanese citizens abducted by North Korea’s agents decades ago. But he has been hawkish on North Korea, championing strong enforcement of sanctions against Pyongyang.

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

Mr. Warmbier, a student at the University of Virginia, was arrested while on a trip to North Korea for stealing a propaganda poster. In 2016 he was sentenced to 15 years hard labor.

More than a year later he was released and returned to the United States gravely ill, with doctors saying he suffered a catastrophic brain injury. He died in June 2017.

Mr. Trump has taken credit for the return of Mr. Warmbier and a handful of other Americans held in North Korea. In the past, the president has pointed to Mr. Warmbier’s injuries as an example of the Kim regime’s brutality.

But on Thursday, Mr. Trump refused to place any blame on Mr. Kim.

“I don’t believe that he would have allowed that to happen, it just wasn’t to his advantage to happen,” Mr. Trump said. “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.”

—Austin Ramzy

Thursday began without any outward sign that talks were about to break down.

President Trump and the North Korean leader, Kim Jong-un, complimented each other in opening remarks at the Metropole Hotel in Hanoi. Neither predicted any imminent breakthroughs, but their comments did not suggest a summit meeting that would end early, abruptly, and without anything to show for it.

“I can’t speak necessarily for today, but I can say a little bit longer term and over a period of time I know we are going to have a fantastic success with respect to chairman Kim and North Korea,” Mr. he said.