The nursing home does not take ownership of the house. However, in most instances – there are some exceptions – your father cannot transfer the house to you.
Instead, your father can keep the house and qualify for Medicaid to pay for his care, but the Medicaid agency will have a claim for reimbursement of its costs when your father dies or the house is sold during his life.
A local elder law attorney can advise you on whether there are any strategies available to you to avoid that claim. You can find one near you here.