RICHMOND, Va.—A political crisis engulfing top officials in Virginia deepened Monday as state lawmakers renewed their demand that Gov. Ralph Northam resign and as Lt. Gov. Justin Fairfax denied allegations that he sexually assaulted a woman in 2004.

The allegations against Mr. Fairfax came as Mr. Northam attempts to survive the discovery of a photo of two people—one in blackface and the other in a Ku Klux Klan outfit—in his 1984 medical-school yearbook page. Mr. Northam, a Democrat, at first apologized for appearing in the...