Tim Cook sat down more than a year ago to watch Apple Inc.’s first scripted drama, “Vital Signs,” and was troubled by what he saw. The show, a dark, semi-biographical tale of hip hop artist Dr. Dre, featured characters doing lines of cocaine, an extended orgy in a mansion and drawn guns.

It’s too violent, Mr. Cook told Apple Music executive Jimmy Iovine, said people familiar with Apple’s entertainment plans. Apple can’t show this.

Across...