The New Orleans-based federal appeals court set to hear arguments July 9 in the legal fight over Obamacare denied Republicans’ request to delay arguments in the case.

The U.S. Court of Appeals for the Fifth Circuit gave the coalition of conservative states challenging the law a few extra days to file an additional brief on whether the Democratic-led states and the House of Representatives have standing in the case. That question is whether they can appeal a district court ruling that declared the entire Affordable Care Act unconstitutional.

In a July 2 order, the court said the states now...