Based on the described facts in your question, my reply is:
If you and your brother are unable to agree to the best situation for the care of your father, the mental capacity of your father is the primary issue. It appears that the nursing home has determined that your father lacks legal capacity to make his own decisions, and that under the POA, your brother has a legal right to make such decisions. A person who lacks legal capacity cannot sign a new POA granting such decision rights to another. A probate court can determine the lack of legal capacity and appoint a guardian (you, your brother, or another person) who would then make such decisions. If your father "just walks out," the nursing home may feel compelled to initiate such probate court proceedings. You may want to discuss this with your brother and the option to petition the probate court for such determination.