Republicans have spent hours this week trying to portray Judge Ketanji Brown Jackson as an extremist on issues of race and an apologist for child sexual abusers. Late Tuesday, Senator Marsha Blackburn, Republican of Tennessee, added another social issue to the list of cultural grievances the G.O.P. is foisting upon her in her confirmation hearings: gender — specifically, what makes a woman a woman.

Transgender rights are dominating outrage on the right, after the Republican governors of Indiana and Utah vetoed bans on the participation of biological males in women’s sports in recent days and a transgender swimmer, Lia Thomas, won a collegiate swimming championship.

Ms. Blackburn brought that swimmer up late Tuesday as Judge Jackson’s first day of questioning was coming to a close, then asked, “Can you provide a definition for the word ‘woman’?”

As she and other judicial nominees before her have done regularly, Judge Jackson dodged the question as outside of her role on the bench, saying, “I’m not a biologist.”

But Ms. Blackburn pressed the point — and the hot-button issue.

“The fact that you can’t give me a straight answer about something as fundamental as what a woman is underscores the dangers of the kind of progressive education that we are hearing about,” she said. “Just last week, an entire generation of young girls watched as our taxpayer-funded institutions permitted a biological man to compete and be a biological woman in the N.C.A.A. swimming championships. What message do you think this sends to girls who aspire to compete and win in sports at the highest levels?”

Social issues that are animating Republican voters, particularly issues of race, are also dominating Republican questioning of Judge Jackson, the first Black woman to be put forward for the Supreme Court. Ms. Blackburn’s insertion of gender into the hearings was not connected to any particular case coming before the Supreme Court, or decided by the high court in recent years, as is tradition at such hearings. A discussion of stare decisis this was not.

But it was connected to the politics of the looming midterm elections. At 11 p.m. Tuesday night, the senator’s office blasted out an email with the headline, “BREAKING: Judge Jackson Cannot Define The Word ‘Woman.’”

As they did all day on Tuesday, operatives at the Republican National Committee grabbed on to the latest line of social attack on Wednesday morning to amplify Ms. Blackburn’s message.