I think we need to make a distinction between 'refusing to provide a service' and 'forcing their beliefs' on someone.
If I say "No, I am opposed to abortions on religious grounds, so I will not provide you with one," I am not forcing you to be religious. I am not forcing you to partake of my religion or worship my God. I am making decision about what *I* will do or not do, based on *my* beliefs. I am not forcing *anything* on you.
And let's face it, everybody operates in accordance with their own beliefs, regardless of what motivates them. Acting based on our own beliefs is not 'forcing our beliefs' on others. It's merely acting in accordance with our own beliefs.
And since you seem determined to claim that religious hospitals refuse to provide services that 'protect' and 'help' people, what are those services? Please be specific.
Er, no, I didn't claim that. Quite the opposite. I said (or meant, anyway), actually, that all the religious hospitals I have experience with did anything another hospital would do regardless of their specific religious orientation.
Would you give a pass to a cop who ticketed people of different races more heavily? That's acting in accordance with his beliefs. Would I be right, if I was a school teacher, to refuse to teach Muslim kids because I believe they're dangerous? (Of course I don't believe that) Acting in accordance with my own beliefs. It's not right, sorry, especially for a hospital.
And yes, actually, as a medical professional, hospital director, whatever, if you refuse a treatment and your refusal directly impacts a patient's quality of life, that is forcing your beliefs on them. Just like a teacher refusing to teach evolution.
If you still don't like my phrasing, how about this: I absolutely do not think that a religious organization should be permitted to allow their specific beliefs to affect medical treatments if that will compromise their patient's health. On the same subject, as stupid as I personally think the idea is, I vehemently believe a Jehovah's Witness patient should be allowed to refuse a blood transfusion for themselves. I just do not believe a Jehovah's Witness surgeon, doctor or hospital should have the right to make the same decision for a patient (Again, hypothetical example.)