A Kentucky county clerk who opposes same-sex unions for religious reasons again refused to issue marriage licenses to any couples on Tuesday, opening herself to potential fines or jail time after the U.S. Supreme Court declined to consider her petition a day earlier.

In response, a federal judge ordered Kim Davis, the Rowan County clerk, to appear in court Thursday concerning a motion to hold her in contempt of his earlier order that she resume issuing licenses. The judge could impose a range of sanctions, including incarceration....