You know, I just looked it up. And the first interracial kiss (or sign of affection in any event) on Star Trek wasn't between William Shatner and Nichelle Nichols in Plato's Stepchildren November 22, 1968. I argue it was in Mirror, Mirror which aired October 6, 1967. The evil duplicate of Lieutenant Sulu (played by George Takei) is distracted by Uhura (played by Nichelle Nichols) to help her crewmates in the other part of the ship. So in that scene, Uhura walks up to Lt. Sulu in a sexy manner and tells him she is supposed to protest and then he's supposed to come back, when he comes onto her like that. Now get this, Sulu puts his arms around her and tells her she's making sense now. Sulu is even kissing Uhura's neck at one point. While he's kissing her, the beep on his console from Scott's work is detected, but goes unnoticed. When the beep stops, she smiles, backs away and then slaps Sulu back into his seat, telling him she's changed her mind again, etc., etc. I'd argue that was the first interracial romance or flirting scene in Star Trek TOS. Anyone disagree?
Yes that is the first true interracial kiss on TV, but since it didn't involve white people it doesn't count on white racist score.sheets. And since William Shatner is Jewish, it probably doesn't count among the anti-semites. Among Asians who can be very racist as well, Takei's kiss would be criticized in Japan.