It's a very old joke that really has no bearing on the contemporary gay community.
In the 1950s and 1960s, pre-Stonewall, a large percentage of gay men identified culturally through Broadway, Hollywood and popular song. (That also corresponded to the time when Broadway provided more material for Hollywood and popular song.) To be gay was SUPPOSED to be to talk about Judy and Marilyn and Bette and Joan and Ethel and Mary and all their vehicles and the ups and downs of their personal lives.
But this was an exaggeration even then. Lesbians were never as obsessed by pop culture as gay men were, and there plenty of gay men who came out of the military in WW2 and Korea who were never involved with or obsessed with the arts.
Even more to the point, since the 1980s and 1990s, the large majority of young gay men coming out know nothing of those camp references, and they probably never will.
So to call the Tony Awards a "gay event" because of the same-sex-spouse thank-yous has some validity; to call it a "gay event" because a significant portion of the relative small audience is gay is probably statistically correct; but to call it a "gay event" because Letterman assumes a majority of gay people watch it is just misguided and misinformed--and, ultimately, homophobic.