If you are going to stop racial tensions, you've got to stop the false accusations, as well. If there were no racism, but there still were false accusations, you would still have racial tensions.
What is the solution, then? Part of the answer is that false accusations should be called out, even as racism itself is. This is a tender thing, for sometimes a person makes an accusation, honestly believing he or she is being discriminated against. While it should be pointed out to them that they are wrong, I don't know that it should go so far as being a rebuke. Treat them tender, for it should not be forbidden to say you believe someone discriminated against you if you honestly believe that. So, when I say they should be called out, I mean only that they should be corrected.
Other times, false accusations of racial bigotry come from those simply trying to make mileage off those accusations. They seek to get their way or hurt others with their false accusations. The rebuking here should be firm.
And, to solve this problem, we should teach ourselves. We should have messages warning how serious it is to make false accusations. Television messages. Radio spots. Internet advertising. The messages should point out the harm done when someone makes a false accusation, how their reputation is tarnished, how they are branded as a racist when they are not one.