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Donald J. Trump kept his vow to remain unbeholden to political correctness after Thursday night’s debate, continuing to criticize the Fox News anchor Megyn Kelly for asking him questions that were too tough while he was on stage.

Mr. Trump promoted tweets that called Ms. Kelly a “bimbo” and that said she “bombed” as a debate moderator. The feuding fed into lingering accusations of chauvinistic behavior by the billionaire.

Ms. Kelly infuriated Mr. Trump when she suggested that he lacked the temperament to be president because of offensive remarks he had made about women. She pointed out that he had referred to them as “fat pigs, dogs, slobs, and disgusting animals.”

When Mr. Trump joked that he was only talking about his nemesis, the television star Rosie O’Donnell, Ms. Kelly said that the issue was more widespread.

“Your Twitter account has several disparaging comments about women’s looks,” she said. “You once told a contestant on ‘Celebrity Apprentice’ it would be a pretty picture to see her on her knees.”

Mr. Trump told MSNBC on Friday that he did not recall many of the references Ms. Kelly was making and that, “We’re going to have it checked out.”

But the exchange highlighted the idea that Mr. Trump had been sexist over the years. The Times reported last month that he once stormed out of a deposition because a female lawyer needed to pump breast milk. He called her “disgusting.”

Mr. Trump had an ongoing feud with Ms. O’Donnell, whose appearance he regularly ridiculed. He has also mocked Cher for her plastic surgery.

While Mr. Trump has become increasingly popular among a wide range of Republicans, he continues to struggle with women, polls show. A July Quinnipiac poll found that 37 percent of Republican women would never support him.

Absolutely disgusted by 1) Donald Trump's bigoted responses about women and 2) the roar of cheers and applause in support of him. — Taylor Yeagle (@TaylorKinsey) August 7, 2015

If more voters adopt that view, it could be a headwind that Mr. Trump’s campaign cannot overcome.