It's an anachronism, there is no such film
The City on the Edge of Forever is set in the year 1930 (emphasis mine):
Kirk: [reading from a makeshift tricorder] February 23rd, 1936. Six years from now. The President and Edith Keeler conferred for some time today
Star Trek Season 1 Episode 28: "The City on the Edge of Forever"
However, Clark Gable simply wasn't a household name in 1930. According to his Biography.com page his breakout role was The Painted Desert, which was released in March 1931 and, according to The Film Daily (the main source for entertainment news at the time), still in production in December of 1930:
Activity at Pathé is about at its highest rate since E. B. Derr became production chief. Two specials starring Bill Boyd, "The Painted Desert" and "Beyond Victory" are being edited.
"6 features and 6 shorts in the works at Pathé" The Film Daily December 7, 1930
If you look at Gable's filmography, he doesn't even start getting his name on the poster until Laughing Sinners, released in May 1931, and doesn't get top billing until August's Sporting Blood. Although he did appear in films released in 1930, he was an uncredited extra and it's doubtful that Edith would have known about him.
Note that Gable's name was a last-minute addition to the episode; the shooting script (uploaded by a fan), which is dated January 1967, has Edith mentioning Richard Dix, a far more period-appropriate reference:
Edith: If we hurry, we can catch that Richard Dix movie at the Orpheum. I'd really like to see it, Jim...
Star Trek Season 1 Episode 28: "The City on the Edge of Forever" (shooting script)
According to his IMDB page, Dix appeared in two movies released in 1930: