Yes nursing home patients are eligible for a stimulus check.
As long as they have applied for the stimulus check or someone has applied for the stimulus check for them then they should get the stimulus check.
My Grandma got a Stimulus Check and she's a nursing home resident and I used the stimulus check to help buy her things she needed and clothes.
The nursing home should not get the stimulus check because the stimulus check is meant for the nursing home patient.
If the nursing home patient got a stimulus check before then if the new stimulus check is approved then they should get another one.
I'm hoping that we do get another stimulus check soon because it will help pay off some bills.