Catholic hospitals provide about 25% of hospital care. In some areas it is much greater. They also provide a greater proportion of care to Medicaid and the uninsured than other hospitals. I do hope the Bishops stand firm and we will see who blinks first. If Catholic hospitals shut down, private entities will not have the resources to buy many if not most of them because, hospital margins are small, and Catholic hospitals also rely on charitable enowments. The government could try to take them over, but this would be worse than bailing out the car companies. Just think how bad the access would be if the Catholic hospitals closed down.
The only thing that might make them stop before that point is if they are just doing it for votes--they may feel they have shown enough zeal to please their base and they were forced to give in to the bad guys (the people who cling to their God or their guns). But it could be that this is so central to their thinking that they can't compromise even if it hurts them in the election.