I think you have the freedom to choose who provides your medical treatments. But, I also believe that you don't have the right to demand a change of personnel at that particular medical facility based on anything other than performance. In other words, I believe the proper response from the medical facility should be "this is who we have available for you. Unless she has done something wrong, she will remain. If you choose to leave, you have that option".
I have been in the situation where I demanded a change in nursing. It wasn't because of race though. In fact, there were two instances and one of them, I have no idea...don't remember the nurse's skin color. The second one, I do.
But, both of those nurses made fairly severe mistakes in what they were supposed to be doing. I was in the hospital for an extended period, I knew exactly what was written in their work instructions. Both times, even with myself and a family member telling them they were not following the written instructions, they continued. The first came back later, in quite a rush, and made an attempt to correct the mistake. The second occurrence was pretty rough actually. I finally stopped her half way through and demanded a different nurse immediately. I never saw either nurse again.