If confirmed by the Senate, Mr. Gaetz, 42, will lead a department that carried out a sex-trafficking investigation into him.

The Justice Department ultimately declined to charge Mr. Gaetz with any wrongdoing after a two-year investigation.

But Mr. Gaetz remains the subject of an ongoing House Ethics Committee investigation into whether he engaged in sexual misconduct, accepted improper gifts, dispensed special privileges or engaged in illicit drug use.

He has denied any wrongdoing, including ever having paid for sex with a minor or paying for sex. In September, Mr. Gaetz said that he would no longer voluntarily participate in the panel’s probe.