We consider content delivery over fading broadcast channels. A server wants to transmit K files to K users, each equipped with a cache of finite size. Using the coded caching scheme of Maddah-Ali and Niesen, we design an opportunistic delivery scheme where the long-term sum content delivery rate scales with the number of users in the system. The proposed delivery scheme combines superposition coding together with appropriate power allocation across sub-files intended to different subsets of users. We analyze the long- term average sum content delivery rate achieved by two special cases of our scheme: 1) a selection scheme that chooses the subset of users with the largest weighted rate, and 2) a baseline scheme that transmits to all K users using the scheme of Maddah-Ali and Niesen. We prove that coded caching with appropriate user selection is scalable since it yields a linear increase of the average sum content delivery rate.