Will the Google Play Game Services multiplayer allow (turn-by-turn) gameplay like DrawSomething?
You are not talking about asynchronous gameplay (which is explained here).
You are talking about turn-based gameplay. But I think you can relinquish the idea of using Google Game Services, because of the missing realtime gameplay (= you don't need to establish a connection between the clients).
I would recommend using a database, which contains everything relating to the game like the gamestate, the players, who did the last turn and things like this.
If you can do synchronous gameplay (i.e. playing together in realtime) you can always do asynchronous gameplay as well. The only deciding factor is whether you're able to establish the required connection at any time.