Was Cary Grant the first actor to say "gay" onscreen?

It's indeed most likely this was the first instance of this usage. Plenty of sources support this theory.

The term Gay had definitely become a sexualized term by the 17th century to mean "uninhibited by moral constraints" and was then attached to homosexuality by the early 1900s. But it's widely believed this usage had not been seen in a Hollywood film before Bringing Up Baby, and it probably only saw the light of day because Grant improvised it:

Grant's explanation for wearing women's clothes in the film, "I just went gay all of a sudden," was improvised on set, which may explain how it slipped by the Production Code Administration (PCA), Hollywood's self-censorship group. Between 1934, when the PCA began strict Code enforcement, and 1961, when the Code was amended, any mention of homosexuality was strictly forbidden on screen. This marks the only use of "gay" to mean "homosexual" in a Hollywood film of that era. Some historians have suggested that it's the screen's first use of "gay" in a sexual context.

The usage not being exactly common knowledge yet also helped bring about this seminal moment:

Since this was a mainstream movie at a time when the use of the word to mean homosexual would be unfamiliar to most film-goers, the line could also be interpreted to mean, “I just decided to do something frivolous.”

As for crossdressing for comedic purposes, it was not the first time. Buster Keaton and Stan Laurel donned women's clothes as early as 1924.

[As a point of interest, a reliance on lingual obscurity helped writers slip another gay term past censors in The Maltese Falcon 3 years later by describing Wilmer, a character considered by many to be gay, as a 'gunsel', a Yiddish word for a young homosexual companion:]

The novel was originally serialized in a magazine, Black Mask, whose editor refused to allow vulgarities. Hammett used the word gunsel knowing that the editor would likely misunderstand it as relating to gun, and therefore allow it.