The U.S. Supreme Court June 25 denied two petitions challenging an appeals court ruling that upheld Obama-era Labor Department regulations on tip income.

The high court’s move means a ruling by the U.S. Court of Appeals for the Ninth Circuit remains valid. The Ninth Circuit is the largest of the regional federal appeals courts and establishes precedent for federal courts in 11 Western states and territories.

The Labor Department changed its position during the course of litigation. The agency put...