How do I deal with angry family members that are wrong?

Does this care home not have a supervisor or Administrator. If so, this person should be brought to their attention and maybe they can tell him if his attitude doesn't change, then he will be banned from the facility. That he is disrupting the other residents. Actually, if he has any complaints he should be talking to the person who oversees the facility. Not the staff.

Follow the chain of command at your own facility. When this person visits keep a careful diary record of what is said and done, of accusations. Report accusations of stealing, especially, to your superiors. As to angry and loud, much of this, and often enough, it can be diffused by listening carefully for a while. Ask your superiors who best to direct him to. Supply him with phone numbers of the chain of command at your facility and tell him "I know that so-and-so can assist you with this problem. Here is his/her phone number. I know he/she will want to help you with this." Always greet by name when you see him with a start of "Mr so and so, good to see you; how are you today". Just honestly, no matter what occurs take it seriously, and try to deflect it off to those who "make the big bucks" to handle these situations. Just give them the old "I wish I could take care of this for you, but I can't. I can refer you to _________ though, and I know they will help you." Might be upsetting to others, you are right; at some point he may need to be counseled about how to visit in order to keep visiting privileges, but that is most dependent on the excellent records you keep. And you would be surprised, half the onlookers probably look on it as "today's entertainment". My brother is in assisted living and just delights in telling me "the stories".

Have you talked with your supervisor about this person's behavior?