Keep in mind, I posted a link to Wiki, because it's convenient. But, it's convenience doesn't necessarily mean it tells the complete picture. To paint a complete picture with respect to "choice of law", I'd need to provide several links to sites that are shielded behind a pay-wall.
There are dozens and dozens of precedents established for "choice of law" provisions to be based on religious canon law - Jewish law, Catholic law and even sharia law. It (jurisdiction) doesn't necessarily need to be limited to a physical jurisdiction, despite how it's described in Wiki. It can reference a body of law, like a religious canon.
In the interest of brevity (and accuracy) I would point you to several discussions at Volokh Conspiracy. Those links may be found below, and they include discussions about when such application is appropriate (like in a binding arbitration agreement), and when it wouldn't be, like the child-custody case that's discussed. The learned professor, Eugene Volokh, does a much better job of explaining why this sharia business is overblown and irrational.
Why American Courts Should Sometimes Consider Islamic Court Rulings (and Islamic Law)
American Court Refuses to Honor Lebanese Islamic Court Child Custody Order
Court Rejects Claim that AIG’s Use of Sharia-Compliant Financing Violates the Establishment Clause
From the first link: the blog post refers to interpretation past acts that occurred in foreign jurisdictions, e.g., marriage, divorce, adoption. It does not refer to new acts taking place within the United States, but using the laws of a foreign jurisdiction. Is this correct?
The second link, to me, is essentially similar. Despite the fact that the father took the children to Jordan under false pretenses (a religious divorce), he instead filed for custody in Jordanian Sharia court once in Jordan with his children. The wife traveled back to Massachusetts and filed a case in Massachusetts. Again, an act taking place in a foreign jurisdiction, not in the United States under foreign law.
Regarding the third link, I'm not sure of the relevance. That link tells of the government being or not being forbidden to invest in companies that have some aspect of religious endorsement, such as selling religious wines or clothing or books, when that religious activity is not primary to the business (such as a supermarket selling kosher food amongst all its food offerings). Investing in Ralphs does not endorse Judaism because Ralphs sells Manischewitz. What is the significance of this to allowing Sharia Law to determine contract compliance?
Regarding the fourth link, I suppose the intent was to read the 196 reader comments and not the advert for the radio show. Admitting that I did not read past 5 or 6 replies, my thought regarding building a western barrier against Sharia Law would be this: how does it comply with the Constitution's Supremacy clause, and the 14th amendment's equal protection for all? Even if somebody agreed to Sharia arbitration, couldn't the whole thing still be challenged by the loser under Article VI, "...and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding." To me, Article VI invalidates any Sharia Court ruling, regardless of whether the parties agreed to it or not, if someone else were to challenge it.
-PJ