A hospital is just a building and the organization that owns the building.
The real question is, should hospitals be allowed to force or forbid doctors from providing medical care?
A doctor (Catholic or not) should never, and can never, be forced to perform a medical procedure, including abortions. And they also shouldn’t be forbidden from performing a medical procedure.
Hospitals just provide rooms and equipment so that doctors can provide the care that their patients need, within their ability to provide that care.
Right, because you would want to be the patient to undergo a procedure by a doctor that never performs that procedure?
No sane country does this.
Maybe go live in a dictatorship if you like to force people to cut into other people’s bodies.