Representative Matt Gaetz, the Florida Republican who is the focus of a Justice Department inquiry over whether he broke sex trafficking laws, maintained his innocence on Friday and insisted he was the victim of “the leaks and the lies” of his critics.

The remarks came during Mr. Gaetz’s first major public appearance since The New York Times reported last week that investigators were said to be examining his involvement with women who were recruited online for sex and given cash payments, as well as whether Mr. Gaetz had sex with a 17-year-old. The investigation developed from an inquiry into a former Florida tax official and friend of Mr. Gaetz, Joel Greenberg.