Is this an attack on religious freedom?
As I understand the flap, the Church is relying on another law, the Religious Freedom Restoration Act, to exempt them from the contraception mandate. In other words, that law theoretically protects all religious groups from any such attack.
Some argument I see in their rags goes like this: some churches may enjoy the exemption, but the Catholic Church will not, because it serves too many citizens at large (regardless of religion) in its hospitals. In other words, the injury to their religious rights gets diluted by the large population of unknown faiths receiving services. Since no polls are taken at intake, they can't prove their numbers and claim the exemption.
Also note, the Dubbya faith-based initiative stuff pushed more public services into the religious sector, followed by funding to incite the groups like the Catholic Church to rush in and throw more of their capacity at the demand.
So (I am assuming from what I've read) it's partly just an attack on the funded programs they went after. They can pull out, but that would wreak havoc on patients who rely on them. If they completely privatize, then they could (theoretically) require a baptismal certificate at intake, which would probably get them their exemption, but at the expense of Non-Catholic patients.
It seems to drag in a lot of old unresolved issues. So I don't think it's just the simple clash between two forces over this one basic issue, as some of the folks here will say.