Let me start out by saying that I don't like the health care reform legislation, simply b/c it does little to reduce costs in the long term. The AMA monopoly on medical care is the main reason why costs are so high, and the incentive structure for health care providers is out of whack in a way that encourages people to get unnecessary care and pays doctors exorbitant amounts to provide it. The AMA is powerful, doctors think they deserve it, and thus the system stays in place.
Let's separate this discussion from our feelings about the merits of the health care legislation as a whole. The reality of this situation is that ND is making a political move to advance a socially conservative agenda, nothing more. Institutions like Notre Dame constantly provide funds to states, and thus indirectly "support" or "pay for" things like the death penalty, which runs counter to Catholic beliefs. For instance, even though they are tax exempt, they pay payroll taxes and provide payments to localities and states in lieu of property and income taxes. Under the logic being espoused by a whole bunch of members of this board, they should not have to provide any money to the state if the policies of the state are in conflict with the religious doctrines of the university. This is wrong, this is not how institutions work in the US. Institutions are required to comply with federal law. ND has to pay payroll taxes to the state that allows the killing of criminals, despite the fact that this runs against Catholic doctrine.
ND is currently receiving enormous benefits from operating under the law of the federal government, including the ability to raise money and own land without being taxed. As an employer and a nonprofit organization, they are subject to federal law. The fact that the government is providing an exemption that would mean the insurers provide contraception to employees is silly, but it's a token gesture meant to allow the university to maintain that it is not taking any action that might directly or indirectly lead to activity that is inconsistent with its own doctrines. This is satisfactory to many Catholic groups, but not Notre Dame.
Rhode Irish is right that the only reason this has an even remote chance of going forward in court is that the Supreme Court is advancing a radical right wing agenda. It's extremely embarrassing to me that ND is playing any part in it. There are lots of things about the Catholic church as an institution that I value tremendously - its backward positions on gender and sex are the ones that make me truly ashamed. There's a sex issue that the Catholic church has to deal with, and this isn't it.