This is almost certainly an anachronism.
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's first credited appearance was in The Painted Desert, which was released in 1931. Although he did appear in films in 1930, he was uncredited and it's extremely unlikely Edith would have known he was in them.
Although The Painted Desert was filmed in 1930, it's exceedingly unlikely that it would have been in theatres in that year; according to The Film Daily, the main source for entertainment news at the time, the film was still being edited 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
Note that, according to Memory Alpha, Gable's name was a last-minute addition to the episode; the shooting script has Edith mentioning Richard Dix, a far more period-appropriate reference. According to his IMDB page, Dix appeared in two movies released in 1930: