I'm not entirely sure I understand what you mean by this...we're a nation largely still comprised of Christians, but we aren't a Christian nation. That is, our government is one willfully created to protect religious right while maintaining no particular religious institution as its wheelhouse.
So Muslims who can accept our process and form are as welcomed and protected as any Jew, Hindu, Christian, etc.
Sharia law in terms of domestic, civil matters, provided it functions within the framework of our law, is no more impermissible than Hasidic courts maintained by Jews. In fact, so long as they are subject to our civil authority and precedent they can be an aid, by reducing the drain on secular courts.
The religion protected may have been the Christian Religions, Denominations basically.
Hinduism is something else, but Jewish and Christian courts, we have the Torah and the Bible. With Islam or Muslims there would have to be another book.