After all, Mr. Kasich has not won a single state aside from Ohio. He has fewer delegates than Senator Marco Rubio of Florida, who ended his own bid more than a month ago. And on Tuesday, Mr. Kasich even lost to Donald J. Trump in McKees Rocks, the Pennsylvania community where he grew up.

“I said, ‘What do you think, sweetie? I’m inclined to keep going,’ ” Mr. Kasich recounted at a town-hall-style event on Thursday.

By Mr. Kasich’s telling, his wife, Karen, agreed.

“She said: ‘The people need a choice. And if you don’t give them a choice, who will?’” he told the crowd, which broke out in applause.

“So I’ve decided to keep going,” Mr. Kasich said. “And there are going to be people who are going to criticize me for that. And it’s not always an easy road. I’m going to do my very best.”