A federal appeals court ruled that the Consumer Financial Protection Bureau’s requirement that its director be fired only for cause is constitutional, dealing a blow to bureau opponents seeking to get the question before the U.S. Supreme Court.

Provisions in the 2010 Dodd-Frank Act that gave the CFPB independent funding and a single director that could only be fired for cause met constitutional muster based on existing Supreme Court precedent, a three-judge panel of the U.S. Court of Appeals for...