“How can you? That’s not .... You can’t do that,” he said.

Mr. Colbert repeatedly seemed to try to prod Mr. Biden to run, saying at one point that the American people were inspired by the vice president because of the tragedies he had endured. In addition to the death of his son, Mr. Biden’s first wife and his daughter were killed in a car accident in 1972.

Mr. Colbert, who lost his father and two brothers in an airplane crash, told Mr. Biden: “It’s going to be emotional for a lot of people if you don’t run. Your example of suffering and service is something that would be sorely missed in the race.”

Mr. Biden seemed almost embarrassed by the praise, saying: “I feel self-conscious. The loss is serious and it’s consequential, but there are so many other people going through this.”

At Mr. Colbert’s urging, he also discussed religion and philosophy at length, describing his Roman Catholic faith as providing an “enormous sense of solace.” He attributed part of it to the rituals of his religion.

“I go to Mass, and I’m able to be just alone, even in a crowd,” Mr. Biden said. He recalled how his wife, Jill, once taped a note on his bathroom mirror with a quote from the philosopher Kierkegaard that said, “Faith sees best in the dark.”

There was one moment when Mr. Biden seemed to suggest where he might find the strength to run. He noted that his mother had a favorite expression: “As long as you are alive, you have an obligation to strive, and you’re not dead until you see the face of God.”