Representative Jason Chaffetz of Utah withdrew support from Mr. Trump by telling a local television station, “My wife and I, we have a 15-year-old daughter, and if I can’t look her in the eye and tell her these things, I can’t endorse this person.”

But some activists, experts and party strategists say that display of empathy illustrates that the lives of minorities are not as valued as those of white women.

Hours after the release of the video, Symone Sanders, the former national press secretary for the presidential campaign of Senator Bernie Sanders of Vermont, took to Twitter: “You apparently can say whatever you want about Mexicans, Hispanics & Black people, but the Republican Party draws the line on white women,” she said on Friday night.

In an interview on Saturday, Ms. Sanders said many Republicans asking for Mr. Trump to withdraw from the race and referring to their families were illustrating that they could not empathize when others were targeted.

“We are still measuring folks by standards of whiteness, and it took for Donald Trump to essentially speak poorly about white women in this country en masse — it took Donald Trump to do something and say something where white people could put themselves in that position to say, ‘Whoa, whoa, whoa, whoa. You’ve gone too far, buddy. This is too much,’” she said.