“Why,” she asked bluntly, “should the women of America vote for you over our sisters who are kind of more qualified?”

Mr. Buttigieg responded as he has to past questions about his comparative lack of experience: by emphasizing the “executive experience” of being a mayor. But it was not the most forceful of statements, and he ended by saying that if some Democrats were set on voting for a woman, he understood.

“Whether you decide to be for me or not, I promise that I will be for you,” he said.

In one moment that quickly drew criticism on social media, Mr. Buttigieg — asked whether it had been appropriate for fellow Democrats to push Senator Al Franken to resign after eight women accused Mr. Franken of sexual misconduct — said, “I think it was his decision to make, but I think the way we basically held him to a higher standard than the G.O.P. does their people has been used against us.”

When Mr. Matthews pushed, he repeated, “It was his decision.” Pressed further, he said, “Well, it’s not a bad thing that we hold ourselves to a higher standard.” Asked a fourth time, he finally said, “I would not have applied that amount of pressure at that time before we knew more.”

The event ended, however, just as genially as it began. Mr. Matthews played a video clip of a young Mr. Buttigieg asking a question of Richard A. Gephardt, the former House Democratic leader, during Mr. Gephardt’s 2004 presidential campaign: Why, he asked, was Mr. Gephardt the only presidential candidate not attending a Rock the Vote event? Did he care about young voters?

Mr. Buttigieg, who seemed taken aback by the clip, said with a laugh that he felt bad about how harsh he had been. Then Mr. Gephardt himself appeared on screen to turn the tables — except his question was the softest softball of the night.

“I’ve always been optimistic about America because the people are good and they’re good citizens,” he said, before assuring Mr. Buttigieg that he was doing a “great” job on the trail. “You’re out there now meeting thousands of them. Am I still right?”