One right-wing commentator, Dinesh D’Souza, appeared on Fox News to question whether Ms. Harris, the junior senator from California and a child of immigrants from Jamaica and India, could truly claim she was Black. And on Tuesday night, Tucker Carlson, the Fox News host, mispronounced her first name, even growing angry when corrected.

“So what?” he said, when a guest told him it was pronounced “Comma-la.” (Fox News declined to comment on the exchange.)

On Twitter, Eric Trump, one of the president’s sons, favorited a tweet, which was later deleted, that referred to Ms. Harris as a “whorendous pick.” Jenna Ellis, a senior legal adviser to the Trump campaign, posted during Ms. Harris’s first speech as Mr. Biden’s running mate on Wednesday, “Kamala sounds like Marge Simpson.”

Mr. Trump added to the barrage with a racist tweet on Wednesday morning claiming that Mr. Biden would put another Black leader, Senator Cory Booker of New Jersey, in charge of low-income housing in the suburbs. That tweet did not mention Ms. Harris, but it continued Mr. Trump’s tactic of playing into white racist fears about integration efforts as he declared, “The ‘suburban housewife’ will be voting for me.”