I don't believe I've read in any patient's rights manual that patients/residents have the right to only accept staff members of a certain race.
However, the resident/employee issue really goes hand-in-hand. If the resident does not want a staff member of a certain race, it's probably due to prejudice/racism. Therefore it is unfair to subject your staff member to caring for a patient who will likely be rude/derogatory toward the employee.
For that reason, (while we hate to act like these sort of requests are acceptable) it's probably best to accommodate the resident when possible and try to assign staff accordingly. However, the charge nurse or DON should address this directly with the resident and say something like, "Mr. Jones, all of the staff here a X Facility are knowledgeable, kind, skilled and competent. While we do not approve of requests for certain staff based on race, we feel that the best decision for our staff is to accommodate you at this time. However, you need to know that there may be times when this is impossible, and if you wish to receive care here, you will need to accept the staff member that you are assigned. No matter who is caring for you, you will be treated with dignity and respect, and we expect the same courtesy to be returned to all our staff."