The issue of religious liberty is about all religious denominations and religious individuals having the right not to violate their own belief systems.
To the extent that this is true, it is the religion's own duty to make that possible, and not in any capacity a political authority's.
Why? Because there is no way for a political representative to even attempt to take such a responsibility without also acquiring and using the power to arbitrate what "should" count as a religion and also what would be within or without the proper boundaries of any specific faith.
That would be both unnecessary, pointless and
very ill-advised. Those matters are anything but trivial, and not at all a legitimate subject for politicians to decide.
If a particular denomination owns and operates a hospital, they have the right not to have to perform abortions if that is against their belief system.
They certainly do... if they decide to let go of the hospital and the duties that come with it.
If they don't, then they are subject to the same moral duties and the same laws as everyone else who owns and operates a hospital - or they should, at the very least.
The woman wanting an abortion can go some place else.
Except that there may not be a proper place in adequate range, and city management is not supposed to allow for violations of basic civil rights in order to protect religiously-motivated arbitrary privileges.
Hospitals are expected to fulfill their roles and the corresponding duties, not to seek loopholes to excuse them from those.
There are plenty of other places available.
Are you trying to argue that only hospitals that do not seek religious privilege should count? Don't you see the logical consequences of that claim?
If a particular individual believes that homosexuality is contrary to their set of beliefs, then they have a right not to associate with or do business with a homosexual for any reason if that is their choice. Again, there are plenty of other places a homosexual can go. There is no need for any religion or person to be forced to go contrary to their will in these matters.
Again, that would be no reason to create unfair, injustified and unworkable religions exceptions.
Instead, that would mean that said individual would better dissociate himself from roles that demand him not to discriminate.