That's a good question. I think if you isolated a specific religious law - like Sharia - it probably would not survive judicial review. I would be confident such a law would be struck unanimously by the Court.
In fact, there's a case (on an Oklahoma statute, I believe) that is working it's way through the federal judiciary. That law has already ruled unconstitutional by the district court, and enjoined. The Circuit Court refused to stay that ruling. It's unlikely that law (it may have been an actual amendment to the state constitution) will survive.
I think it's possible that a more general law, prohibiting the application of any religious canon law in arbitration, has a better chance of surviving judicial review, but it would be far from certain that it would survive.
Places in the US where contracts among Orthodox Jews use the Torah as the law would be very unhappy about abolishing the free exercise of their religion.