If the hospital wants to say no, then let them say no - let the hospital set the policies based on what's logical to make the hospital run well.
What if the hospital were to enforce 'racial' segregation? On the basis that it would run better.
From my perspective, I don't care what a private company does. If they want to segregate, that's fine. I won't associate myself with them, but I'd say that's their right. Obviously this doesn't fit with most of the SCOTUS cases that have happened, so I realize I often have to throw my perspective out the window on stuff like this.
It wasn't the Supreme Court that ended segregation for private companies. It was the Civil Rights Act of 1964. The Supreme Court had historically been opposed to such legislation, ruling the similar 1875 Civil Rights Act unconstitutional.