Its about respect, not the beliefs per se, or anyone else's opinion of them. People's beliefs are sacred to them, and they have a right to hold whatever beliefs their hearts consider important to them....what they don't have is the right to impose those beliefs on others.
Forcing someone to comply with their own conscientious beliefs is dead wrong. (pardon the pun) It is just as wrong as forcing a baker to make a wedding cake for a gay couple when that is against his Christian conscience to do so.
If religious hospitals receive public funding and accept public patients, then they have no more right to demand that certain practices that they consider a violation of their religious beliefs, be imposed on others who are not offended at all. If they became self-funded and independent religious institutions....then, different story.
I guess it's a matter of respect all around. A baker should be able to post a notice in the window of his shop stating that he will not be making cakes specifically for gay weddings, but if they wish to purchase a cake that they can take home and decorate themselves, then no harm done. He doesn't refuse to serve gay people in his shop.
We see notices in shops sometimes saying that patrons should 'not ask for credit because refusal may offend'. Same principle applies here, as far as I can see.
Religiously based hospitals who have objections to certain procedures can also make it known that those procedures will not be performed in their establishments. There are always alternative places to buy a wedding cake or have a medical procedure carried out.....why does there have to be a fight about it? Why are humans so prone to wanting to force their will on others, yet complain when others want to do the same to them?
Free will should be free......and as long as there are alternative options, what does it matter in the big scheme of things if you can't get what you want at one place and have to find another?
Is it really that hard?