Obama's Contraception Mandate Won't Survive Strict Scrutiny
Those who argue that the HHS contraception mandate is legal are wrong because they are looking at only half of the equation. I haven't looked at it as closely as has Richard, but I doubt that the mandate violates the Free Exercise Clause as interpreted by the Supreme Court in Smith v. Employment Division. Smith says that a generally applicable law whose purpose is not to regulate religion does not automatically violate the Free Exercise Clause. Otherwise, the Court worried, religious groups might seek exceptions from the criminal laws, tax law, etc. that just apply to all groups and individuals in the normal course.
But Smith does not settle the matter, because Congress passed the Religious Freedom Restoration Act in 1993. RFRA reversed the Smith approach, and instead restored the previous test, which held that the government could not substantially burden someone's free exercise of religion unless in furtherance of a compelling government interest, using the least restrictive means possible. This is known as strict scrutiny, and few laws survive it.
The Supreme Court struck down RFRA in City of Boerne v. Flores in 1997 but — and this is the important part — only insofar as it applied to state governments. RFRA still applies to the federal government: Congress can require the federal government to be more generous on individual rights than the court requires. I cannot see how a Court would conclude that the contraception mandate survives strict scrutiny — I don't believe the courts have found that the provision of a commercial product like health insurance is a compelling government interest (very few things are — in fact, I think there are only two: war and racial diversity). Thus, those who claim that the contraception mandate does not violate religious freedom are wrong, because they forget that RFRA is still on the books and plainly forbids the contraception mandate.
It could always be the case that Congress, in passing Obamacare, included somewhere in its hundreds of pages a statutory exception for healthcare from RFRA. But if it did, I have yet to hear about such a provision.