Akuperma: I haven’t looked, but my guess is they are claiming it violates the Civil Rights Act or some similar Act, which applies to anyone who provides a public accommodation in interstate commerce. As we know, interstate commerce has no limits, so it applies to any public accomodation.
The same way you can’t have a hotel that is “Whites only”, or “No Jews.”
The better question is if free exercise will trump it. Under Smith, probably not. But Smith is weak.