When our United States first formed, the individual states acknowledged the English Common Law as the basis of law for our courts. (Reception Statutes.) I believe all states but Louisiana followed suit. (Louisiana accepted the Roman Civil Law which later became an issue in harmonizing uniform building codes.)
For instance, in 1850, California adopted the common law of England (Ca. Stats., pg. 219,), so far as not repugnant to or inconsistent with the Constitution of the United States or laws of the State, as the rule of decision in all the courts of the State. (You can find some discussion of this in early water law cases in the West where riparian rights followed the common law and appropriative law followed the Roman Civil/Spanish law. California ended up in a dual system as applies to water because established mineral claims followed appropriative law. )
I imagine Sharia law would have the same status as Canon or ecclesiastic law, which might inform the court as to beliefs and customary practices of devotees but certainly not take precedence over rights and liberties secured by the Constitutions and laws of the United States and individual states.