No, the hospital cannot impose what is essentially a contract term unilaterally. The one possible qualification: I would like to see whatever written documents were already in place - eg, policy manual - which are sometimes deemed to be a contract between you and the employer and which might create some sort of opening for the employer. However, even if there is such a loophole, non-compete clauses must be reasonable and limited to be enforceable.
Be careful! I agree with Mr. Rensch that the hospital probably can't force you to quit your job at the urgent care facility. However, the hospital can decide that it longer wants to employ you because you continue to work at the Urgent Care facility. Effectively, the hospital can make you choose which job you want to keep. I'm sorry that you're in this difficult situation. Good luck.