should hospitals be allowed to force or forbid doctors from providing medical care?

They provide the facilities, which includes administration and legal and billing. So in that regard, they have to have some kind of say, simply because they need to stock the equipment, train the nurses/MAs, and establish standard protocols for a given procedure. Otherwise, how do you contest a medical malpractice claim?

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.

Doctors can and do regularly incur liability if they fail to perform certain necessary medical procedures, particularly in emergency room settings. A doctor that fails to follow protocol can be subject to malpractice. If, for instance, a Christian Scientist doctor refused to provide a blood transfusion to an individual suffering from sever blood loss or a narcotics prohibitionist doctor attempts to do surgery without providing anesthesia, they can get in some serious trouble.

Religious convictions don't override medical protocols. What's at issue is the legality of the protocols as they stand. Can a woman whose health is at risk from pregnancy receive an abortion without the doctors incurring criminal liability?

Right now, it appears that State AGs in prohibitionist states are threatening the licenses and freedoms of doctors who would provide life-saving care. Hospital administrators are acting as intermediaries because the hospital itself would suffer legal liability if staff knowingly permitted/facilitated an illegal procedure.