What do I do if my father doesn't like his home health-care aide?

Some people of your father's generation are still apt to have misguided racial issues just out of habit, especially if they've lived in a community that has little ethnic variation. Some of that thinking may enter in to his complaining, but then when you suggest you get someone else, he realizes he has no grounds for the complaints and realizes that he even likes her.
It seems that you have two choices - keep doing what you are doing and just detach from his complaining, as he may complain about anyone, just because he needs to complain. The other choice is to go ahead and call the agency. Tell the supervisor that while this caregiver is really very good, your dad has times when he doesn't think he likes her, so maybe there could be a switch in personnel. Make it plain that this isn't the caregiver's fault, but your dad's personality. Frankly, I doubt that a change in caregivers will improve things, and you may get someone who truly isn't good. Only you can decide what's worth doing.
Take care of yourself. We know it's rough.
Carol

It could be a personality issue. When I hire caregivers it is always on a "trial basis" at first. This gives your father the chance to see if they are compatible and if he feels comfortable with him or her. As Carol mentioned, it may just be a conflict in "chemistry" and that is the way I would approach it with the agency.
Also, do you know for a fact that this person is not going into places she shouldn't? One of the directors of an agency I used always told us to lock away personal papers and valuables. His philosophy was "better safe than sorry." And it protects both the client and the caregiver.
I only had one experience where I had to change personnel. The first person they sent over would not listen to what Mom wanted. When she went shopping for her, she bought the products that she liked, not the ones on Mom's list. It also got to the point where she was "bullying" her. So I listened to my Mom and had them send over someone else who worked out beautifully. Your father may be changing his tune when you want to replace this caregiver because change seems worse to him than putting up with unprofessional care. I would give him the benefit of the doubt this time, then if his complaints continue, it just may be his pattern.
If you have concerns about people in your father's home, for long periods of time, you may want to invest in a digital recording device that you can access remotely by your computer.
Good luck...