ASHLAND, Ky.—A federal judge jailed a Kentucky clerk on Thursday for contempt after she repeatedly defied his order to issue marriage licenses to gay and straight couples, imposing his authority in the most high-profile challenge to the U.S. Supreme Court ruling legalizing same-sex marriage.

Kim Davis, the Rowan County clerk, had unsuccessfully appealed all the way to the high court to delay an order by the judge that she issue marriage licenses to all qualified couples, gay or heterosexual.

“The...