When Mrs. Clinton finally got to unload what felt like the pent-up frustration of Everywoman, it was powerful. “This is a man who has called women pigs, slobs and dogs, and someone who has said pregnancy is an inconvenience to employers, who has said women don’t deserve equal pay unless they do as good a job as men,” she said. “And one of the worst things he said was about a woman in a beauty contest. He loves beauty contests, supporting them and hanging around them. And he called this woman ‘Miss Piggy.’ Then he called her ‘Miss Housekeeping,’ because she was Latina. Donald, she has a name. Her name is Alicia Machado.”

Mr. Trump blustered, but didn’t deny any of it. Instead, he dug himself in deeper by saying that Rosie O’Donnell, the comedian who was the target of some of those epithets, “deserves it.”

Mr. Trump’s misogyny is unlikely to turn off his core supporters. And his bullying of Mrs. Clinton — as well as his critique of her reversal on the Trans-Pacific Partnership and his remarks on the effect of globalization on jobs — may play well with white men reeling from technological change, job losses and addiction. Amid this upheaval, some have come to believe that when minorities, immigrants and women make gains, it pushes them further behind.

The debate’s clash over gender was telling for both candidates, and it may have helped establish Mrs. Clinton as a standard-bearer for more than Democrats.