Image Mr. Shapiro, the conservative journalist formerly of Breitbart News, defended his debate offer to Ms. Ocasio-Cortez if she would debate him, saying, “discussion and debate are not ‘bad intentions.’” Credit... Kim Raff for The New York Times

In a phone interview on Friday, Mr. Shapiro rejected that there was anything sexist about his offer, saying that he could see himself making a similar offer to an older, male ideological opponent.

“I would offer Bernie Sanders $50,000 to debate, absolutely,” he said, referring to the Vermont senator. “What in the world does this have to do with her being a woman?”

He also criticized Ms. Ocasio-Cortez’s analogy of catcalling, labeling it an “insane” accusation.

“I’m an Orthodox Jew and I have never catcalled a woman in my life,” he said. “I don’t know how they catcall down in Queens, but it must be weird.”

Corbin Trent, a spokesman for Ms. Ocasio-Cortez’s campaign, declined to respond to Mr. Shapiro. “She said her piece and put it out there, and that’s all we have to say,” he said from Hawaii, where Ms. Ocasio-Cortez is campaigning for a fellow democratic socialist and congressional candidate, Kaniela Ing.

This was not the first time that Ms. Ocasio-Cortez had engaged in a much-watched Twitter tussle since she defeated Mr. Crowley, who had been a contender for House speaker if Democrats managed to flip the chamber in November.