Can I ban my brother’s wife from visiting my father at the nursing home?

Is she behaving in a way that is harmful to his health?

If that is the situation, I would have a talk with the actual family member and see if s/he cant get their spouse to behave in a less imflamatory way.

If that doesnt work, perhaps ask that her visits take place in a public area where her behavior can be observed. Then ask the administrative staff for advice on how to handle this relative.

No. You have no authority to do so.

Health Proxy is for you make decisions regarding his medical needs, if, he cannot do so. Has nothing to do with who visits him.

Why do you want to do this? Unless you have a good reason, it is not decent behaviour.