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There are plenty of laws on the books to defend constitutional rights against religious claims.
The unique thing here is its focus on foreign religious law (read sharia). I bet apoplexy over the hijab is behind this.
Case in point:
http://www.masslive.com/news/index.ssf/2013/08/muslim_woman_asked_to_remove_h.html
The first is reflected in http://www.economist.com/node/17249634 . For purposes of arbitration other legal systems can be used in arbitration.
In the parts of the US that I'm familiar with, this kind of thing doesn't happen. Instead, if there's a clash between civil and religious views, they run in parallel. So years ago I knew a Catholic woman who had divorced and believed she couldn't remarry until she had a writ of divorce from Rome. The legal technicalities had been settled years before and her husband had long since remarried. Orthodox Jews have the same kind of system.
This isn't a big problem in most places but there are people how want to see religion- or culture-specific tribunals or arbitration panels set up in the US more widely. In mixed marriages and families religious arbitrarion can suck for the outside member.
The second is more philosophical. There's a strange kind of bias when comparing the US or individual states with other political entities. If we *like* what's done in the other states/countries then not doing the same thing shows we're backwards. If we don't, then we're better. That's true when it comes to group and individual rights, spending decisions, tax rates, all sorts of things. Both sides do it, but since (D) want to change more things (by and large) than conservatives (hence "conservative"
, my subjective impression is that liberals do this more than conservatives.
Courts do this, too. On rare occasion they will lift legal reasoning from another system or they will cite other countries' laws as motivation in their interpretation of US law. Now, if the US law is really clear that can be hard. US law isn't always clear, and often cases are difficult because the law's a bad fit for them. It doesn't happen often, but when it does it's usually because the court wants to justify a given decision--not because no decision is possible.
I seem to remember a supreme court (SCOTUS, maybe) that struck down a dealth penalty law making heavy reference to international norms and laws. Not theUS Constitution, not treaties, but sort of a consensus among other 1st world countries as to what was acceptable and "evolved." IIRC, it also looked at how states had been trending. Of course, I think they law they struck down was helping to reverse the trend and by striking it down they furthered the trend--one that a majority on the court preferred.
Neither are big problems. But lots of laws are there to please constituent anxiety, some are pro-active and seek to stop a problem before it's common.
It's also easy to conflate the two problems so the problem as a "whole" is greater than the sum of its parts.