Even prior to the current political climate, police overreach in hospitals has happened multiple times. Usually involves police arresting a nurse/doctor for not doing what the police want (even if the medical staff are in the right). Medical staff loses pay for the day, patient harm does happen because of involuntary abandonment, and no punishment is metered for the police.

So yeah, doctors and nurses are quite helpless here.

Could ICE be liable for death or injury incurred from deporting a citizen?

Yes, but only if the court will uphold the law. And at that point, the punitive measures can’t bring back life or brain function.