U.S. regulators ordered genetic-testing company 23andMe Inc. to stop marketing its $99 mail-order kit, citing the risk that false results could cause consumers to undergo unnecessary health procedures such as breast-cancer surgery.

The warning to the Google Inc.-backed company from the Food and Drug Administration follows a debate that has grown as hundreds of thousands of people have turned to direct-to-consumer genetic tests for clues about disease risk and ancestry. Proponents say people have a right to direct access to...