Can a hospital force you to stay because you have a bad cough?

I am in a voluntary mental health unit and was due to be released today. This morning I woke up with a bad cough and now they put me in an isolation room and have told me I am to stay.

I’m not sure if they mean I should stay or I legally have to. I tried to ask but instead of answering my question the nurse said, the swab will be done by tomorrow evening.

If I don’t legally have to stay now and the swab comes back as something contagious will I then be forced to stay.

They are also holding against me the idea of a doctor’s note for my workplace. They said if I leave AMA they will not give me a note for my workplace (and my workplace would likely fire me). Originally I was good to go this morning but now they are trying to hold me for a physical concern and not a mental one.

Can they do this?