Yes of course, I was more pondering what should be done in the position where you would be the attending
You can be "the nice guy" and see if you can pawn said pt off to someone on your service that is of more "appropriate" ethnicity. If successful, everybody wins. But that assumes that the pt only cares about the attending.
Since you said the pt doesn't want to be cared for by white "hospital providers," I'll assume that means everyone- attendings, residents, students, nurses, techs, etc. That's...not possible. You can be the "slightly less nice guy" and try to figure out why the pt feels that way and then attempt to mollify said concerns. Sometimes it works, sometimes it doesn't. If it doesn't, you get the administration involved. 99 times out of 100, those patients want to speak with some sort of administrator anyway. They'll get an apology and politely told that the hospital cannot accommodate such requests. If they don't agree, they can leave and take your business elsewhere. No, you cannot be transferred to another facility. However, I'd pay to hear the phone calls between transfer centers and see "dislike of hospital staff ethnicity" on an EMTALA transfer form.
However, if the pt gets rude and/or aggressive about it, the dynamic changes. The proverbial fuse gets shorter and you spend a lot less time trying to play nice. Security and/or PD get called, and out they go. Or better yet, the pt gets arrested for a&b on healthcare workers and is held in custody at the facility for treatment until discharged for booking.
There very well may be other (better?) ways of handling this, but this is how I've seen it play out countless times.
TLDR: Try being nice, ensure them they'll get the best care possible. If unsuccessful, consult administrators, security, PD, etc PRN.