Most likely, yes, the home does have the right to kick him out. To be sure, check the conditions for expulsion in the contract you signed when your dad moved in. Also, take a look at the list of residents' rights, which all nursing homes are required to provide.
If, after reading the contract and residents' rights, you believe the facility is wrongfully threatening your father with eviction, contact your local ombudsman, a government advocate for people living in nursing homes. The Area Agency on Aging can also help you locate your ombudsman, who can go through the home's history of incident reports to help you figure out if the decision to evict your father was justified.
You should know that evictions don't happen very often. A good facility will do all it can before making a call like that. But if you and the nursing home are at loggerheads, bringing in a third party, such as an ombudsman, can lead to a compromise. Keep in mind, though, that from the nursing home's perspective, letting your father stay is a liability issue. If he harms another resident, the home risks legal action. So if you do try to keep your father there, you could have a strained relationship with the staff.
If it looks as though the nursing home does have the right to evict your father and intends to go ahead with it, you'll generally have 30 days' notice to find a new place for him. You'll still have to figure out what's bothering your father, though, or you'll likely face the same problem at the new facility.