Allow me to start by saying that this answer is provided for informational puposes only and should not be construed as legal advice.
While there is insufficient facts to address your specific case and the law varies from state to state, generally a nursing home has no right to withhold your fathers personal income. Unless the nursing home has been designated as the representative payee through Social Security it is unlikely that they have the authority to unilaterally withhold funds from your father. However, if your father has turned over payments to the facility and he owes them those funds as part of a patient responsibility then they may have a valid legal arguement to retain those funds. If the funds are turned over to the facility automatically, through direct deposit or some other automatic means your father may have the right or ability, if legally competent, to have those funds redirected to his personal account.
However, if your father is recieving any government benefits from the Medicaid program which pays for some or all of his nursing home care, be advised that any gifting of his personal funds may result in terminating your father's continued eligibility for these benefits.
Further, if your father has been determined to be legally incapacitated, and a guardian of the property has been appointed, he would not have the authority to direct the use of his personal funds.
Finally, if your father has limited capacity please be aware that attempts to influence him to part with his personal assets may result in an investigation or report of elder abuse or exploitation. You would be welll advised to strongly consider his ability or desire to make these gifts, free from outside influence, before in attempting to engage in any transfer.