Replying to a Twitter post by the progressive political organization Justice Democrats that included a photograph of Morgan Harper, a candidate the group was backing for a United States House seat in Ohio, Mr. Weisman noted that she would be challenging Representative Joyce Beatty, an African-American Democrat.

Ms. Harper quickly replied to Mr. Weisman’s message, telling him, “I am also black.”

To that, Mr. Weisman replied, “@justicedems’s endorsement included a photo,” as if that settled the matter.

Roxane Gay, a contributing opinion writer to The Times since 2015, joined the discussion with a tweet that said, “Any time you think you’re unqualified for a job remember that this guy, telling a black woman she isn’t black because he looked at a picture and can’t see, has one of the most prestigious jobs in America.”

According to screenshots posted by Ms. Gay, Mr. Weisman sent messages to her saying she owed him “an enormous apology.” Ms. Gay made it clear in a subsequent tweet that she strongly disagreed with Mr. Weisman’s demand.

Erica Green, a national education reporter at The Times, said in an interview that she understood why the tweets by Mr. Weisman, who was her editor, had provoked a backlash. She also defended him as a journalist and colleague, based on her experiences working with him on stories about minorities.

“As a black woman, I feel a little bit better that he is in the room,” Ms. Green said.

Mr. Weisman, the author of the 2018 book “(((Semitism))): Being Jewish in America in the Age of Trump,” stepped away from Twitter for a few months in 2016 after becoming a target of online trolls. Before joining The Times, he worked at newspapers including The Baltimore Sun, The Washington Post and The Wall Street Journal.

Last week, The Times drew intense criticism — from readers, some of whom canceled subscriptions; members of its own staff; and prominent politicians — because of a front-page headline that Mr. Baquet described as “credulous.” The headline, “Trump Urges Unity Vs. Racism,” was changed for later editions.