Generally a person does not sue and get damages paid for something that might have happened.
As to what the government can force on religion, I'd say not allowing snake handling is pretty reasonable given the likelihood of someone being injured of killed. But, like it or not, a hospital is a business, even a hospital that is based on a religious foundation. The government really can't change its business model. We all knew the ACLU was just waiting in the grass for a case like this one. As I said before, it will be a landmark case. The sad part of it is that this case is too weak to even be heard. But it will be. Catholic hospitals, even though they follow a business model, are a mission of the Catholic church. I'm not sure how many Catholic hospitals there are in the US. But I know of one in Nashville, and there is one in my hometown here in KY. There are only two hospitals in my hometown. Can you imagine what would happen if the Catholic church decided to shut down every hospital in the US? It could do that. I doubt it will, but that is an option if the government tries to change what services they offer. I think the only foundation for a finding in favor of the plaintiff is that the hospital did not make an effort to transfer the woman to a facility that would do the procedure. And that is not outlandish. We even have an Air Evac Life Team here in western KY. So, a person with a medical emergency that cannot be treated at a hospital in this area can be in Nashville in less than 20 minutes.
I think it is appropriate to remind posters of the First Amendment of the United States Constitution:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
Catholic hospitals'
mission of healing entails their free exercise of religion. I think all the government can legally do is to say, 'if you won't do the job, send the patient to someone who will.'
As for life saving: You might research the different levels of emergency care that exist in US hospitals. A hospital that is not equipped to provide a particular level of care will airlift a patient to another facility. Happens all the time.
Edited to add levels of trauma centers and services provided.
ED Facility Level Coding Guidelines