Admitting he lied to the public after being caught sending a lewd photo of himself in underwear to a college student, Rep. Anthony D. Weiner on Monday acknowledged making “terrible mistakes,” but said he wouldn’t step down from Congress.

Democratic colleagues said the lawmaker from New York should face an ethics investigation to determine whether his behavior broke House rules. Mr. Weiner said they and his constituents are free to draw their own conclusions, but he thinks he can still serve in office.

“I am deeply regretting what I have done, and I am not resigning,” the married, 46-year-old New Yorker said at a news conference in Manhattan, repeatedly calling his actions “destructive” and “inappropriate,” but that it didn’t rise to the level of abuse of his office...