A nursing home can stop a resident from walking out the door and leaving if they are frail enough or have dementia, Alzheimer's disease etc and it would cause harm to the resident for them to leave.
However if the resident does not have dementia or Alzheimer's and has a sound mind then the nursing home cannot keep or stop the resident from walking out the door.
That would be false imprisonment.
As a nursing home resident that is otherwise healthy and has a sound mind and can make your own decisions you have the right to leave the nursing home at anytime.
But if you have dementia, Alzheimer's or cannot think clearly or really care for yourself then the nursing home has a duty to keep you there unless you're signed out for your own safety.
But otherwise you can simply walk out the door and not return but you should let them know that you're leaving the nursing home just so they know that you're okay.