Religious Arbitration of Civil Disputes:

For an example of how this works in the U.S., see Easterly v. Heritage Christian Schools, Inc. (S.D. Ind. Aug. 26, 2009). As the case name suggests, this involved Christian arbitration, but the secular legal principles would be the same for Muslim arbitration as they would be for Jewish arbitration.

Certainly some matters might by law not be subject to arbitration; I believe that's the rule for child custody decisions in many states. Some arbitration provisions may be set aside if there's evidence that they were entered into under threat of violence or some such (though economic need or social pressure is generally not a reason for that, I think, especially since many businesses enter various contracts because of economic need or pressure from business community norms). Some remedies couldn't be implemented through arbitration. Some sorts of procedures -- perhaps including the application of procedural rules that discriminate based on the parties' sex -- might be prohibited even despite the broad deference to arbitration decisions.

But that's true for all forms of arbitration, whether secular or religious. There's certainly nothing alien to American law in having arbitration tribunals apply religious law (or for that matter the laws of foreign countries with legal systems quite different from ours). Nor is there any constitutional authorization for distinguishing Jewish and Christian arbitration from Muslim arbitration based simply on the denominations involved (though again all these arbitrations are subject to various relatively deferential but not toothless secular constraints, which may of course play out differently to the extent that different arbitral bodies operate differently).

A few relevant passages:

Easterly ... argues that “the arbitration provision in the parties’ contract provides for a process by which Ms. Easterly must forego vindication of her substantive rights guaranteed by the ADEA and ADA and Indiana contract law, and instead rely on biblical scripture to define her rights.” Easterly points to the fact that the RPCC [Rules of Procedure for Christian Conciliation] provide that “Conciliators shall take into consideration any state, federal, or local laws that the parties bring to their attention, but the Holy Scriptures (the Bible) shall be the supreme authority governing every aspect of the conciliation process.” While it is true that “a substantive waiver of federally protected civil rights will not be upheld,” Easterly does not explain how this provision constitutes a waiver of her rights under the [Americans with Disabilities Act] and the [Age Discrimination in Employment Act]. [Substantive rights under antidiscrimination law, as opposed to rights under many other kinds of law, generally cannot be waived up front by contract. -EV] The provision requires the arbitrator to take into consideration the applicable law, and Easterly fails to articulate how biblical principles might conflict with that law to her detriment. In the absence of such a showing, the Court declines to find that submission to arbitration under the RPCC will deprive Easterly of her right to vindicate her statutory rights.

Easterly also argues that “[t]he agreement to biblically-based arbitration in Ms. Easterly’s teaching contract cannot be enforced because the processes are structurally biased and procedurally inadequate.” However, Easterly again fails to articulate how this is so. “The Supreme Court has repeatedly counseled that the FAA leaves no room for judicial hostility to arbitration proceedings and that courts should not presume, absent concrete proof to the contrary, that arbitration systems will be unfair or biased.” She further objects to the fact that the RPCC differ from the Indiana Alternative Dispute Resolution Rules with regard to confidentiality. However, again, “under the FAA the parties are free to agree to any governing rules, and the courts will enforce whatever system they choose.” ...

The FAA “is a congressional declaration of a liberal federal policy favoring arbitration agreements,” and “that questions of arbitrability must be addressed with a healthy regard for the federal policy favoring arbitration.” Easterly has pointed to nothing about the arbitration agreement in this case that would override this policy.