Cheri Jacobus, a Republican political strategist, did not think she had done anything out of the ordinary: On a cable television show, she criticized Donald J. Trump for skipping a debate in Iowa in late January and described him as a “bad debater.”

But then Mr. Trump took to Twitter, repeatedly branding Ms. Jacobus as a disappointed job seeker who had begged to work for his campaign and had been rejected. “We said no and she went hostile,” he wrote. “A real dummy!” Mr. Trump’s campaign manager told the same story on MSNBC’s “Morning Joe.”

Mr. Trump’s Twitter followers, who number about six million, piled on. For days, they replied to his posts with demeaning, often sexually charged insults aimed at Ms. Jacobus, including several with altered, vulgar photographs of her face.

“Cheri is a nutcase,” wrote @LegendaryTrump. Another Twitter user, @stockedwood, wrote, “How the hell does a woman like her get on the air???”