The U.S. Supreme Court declined to wade into a high-stakes jurisdictional battle over labor issues at tribal-owned businesses like casinos and hotels.

The justices May 20 kept in place a decision by the U.S. Court of Appeals for the Ninth Circuit that the National Labor Relations Board had the authority to rule that Casino Pauma violated federal labor law by banning workers from distributing union material near the casino. The Pauma Band of Luiseno Mission Indians, which owns and operates...