The reference is from "Omega Glory", an episode written before there was Star Trek as we know it.
Spock: "Kohms? Communists? The parallel is almost too close, Captain. It would mean they fought the war your Earth avoided, and in this case, the Asiatics won and took over this planet."
The most direct interpretation would be that the United States and the Soviet Union never slugged it out on Earth, but did it on Omega IV. This would be consistent with modern Trek where the participants of WWIII are never specified. Yeah, the US seemed to be involved, but only the novels suggest that it was a bipolar conflict or that the opponent would have been anything comparable to Communist Russia or China. Sure, "East" is mentioned once, but in a context where Lily Sloane speaks of "factions".
Doesn't mean Earth didn't have wars. Even armageddon-level ones, including nuclear winters as per "A Matter of Time".