Play button grayed out, when I mouse over it, it says I must be logged in to battle.net to play, the problem is I am logged in.
The launcher can open in offline mode. On the launcher window, go up to the top right and click on your name there. It will show your status and if you’re offline or not.
If you are online and still seeing this issue, I’d recommend restarting the system and logging in again.
If the problem continues, please refer to this thread here.