No, it was not the first
There are a couple of reasons for initial suspicion of this claim.
- Reviews of the episode when it first aired did not comment on the fact that it was the first interracial kiss, nor seem all that bothered by it.
- References to it being the first didn't show up until many years later.
So what was the first?
First ever on TV (Hispanic?/Caucasian)
I Love Lucy (first episode airing October 15, 1951) featured Desi Arnaz, an Hispanic male, and Lucille Ball, a Caucasian woman, frequently kissing each other.
However, Desi Arnaz was often seen as white with Hispanic ancestry, so this may not count.
First ever on TV (African/Caucasian)
The British Film Institute found footage an earlier interracial kiss from the television play You in Your Small Corner, broadcast on 5 June 1962. When footage was discovered in November 2015, it was declared to be the earliest surviving footage of an interracial kiss on TV.
This footage was found as part of investigation for a BFI panel on race and romance, and during the panel it was announced that an even earlier interracial kiss was found. Hot Summer Night, another stage play, aired 1 February 1959 as part of the British anthology series Armchair Theatre. This included a kiss between Jack 'Jacko' Palmer (John Slater) and Nell Palmer (Ruth Dunning), who are African and Caucasian respectively.
First ever on ongoing TV
A July 1964 episode of British soap opera Emergency – Ward 10 had Louise Mahler (portrayed by Joan Hooley, a Jamaican) and Giles Farmer (portrayed by John White, an Englishman) kiss.
First ever on American TV
The Wild Wild West episode "The Night the Dragon Screamed" aired on 14 January 1966 and had Jim West (Robert Conrad) kiss Princess Ching Ling (Pilar Seurat). The former is Caucasian, the latter is a Filipina playing a Chinese princess.
First ever African/Caucasian on American TV
Movin' with Nancy aired on December 11, 1967 an had a kiss between Nancy Sinatra (Caucasian) and Sammy Davis Jr. (African).
First ever with William Shatner!
Believe it or not, Captain Kirk himself has had an interracial kiss on TV before Nichols. He kissed Pilar Seurat (a Filipina) in the Naked City episode "Without Stick or Sword", airing 28 March 1962.
However, Shatner is playing the role of Burmese sailor Maung Tun, believe it or not. So this is not an example of the characters having an interracial kiss.
First ever on Star Trek
Yes, even Star Trek had an interracial kiss before Kirk and Uhura. The episode "Mirror, Mirror", airing 6 October 1967, had Kirk (William Shatner) kiss Marlena Moreau (BarBara Luna), whom All Movie lists as being "of Hungarian-Philippine heritage".
This was over a year before "Plato's Stepchildren" aired on 22 November 1968.
There's definitely some wiggle room for what counts as "interracial", but it's pretty clear that there are a number of examples of interracial kisses before Star Trek, including ones with Caucasian and African actors.
You could say that Kirk and Uhura's kiss is "the first Caucasian/African kiss on Star Trek" or "the first interracial kiss on Star Trek between main characters", but certainly not "the first interracial kiss on TV".