Open the task manager and check that the game is actually not running. Try rebooting.

I've ran into this issue somewhat randomly. And it's not a case of the game still running because i've checked task manager. The way i got around it is adding a shortcut to my desktop for DiabloLauncher.exe located in the game directory. I am able to start the game this way despite the Play icon being greyed out in GOG Galaxy.

I had googled the issue and it seems to be a problem that others have had with other GOG games aswell. Some people suggested it had to do with administrator privileges in certain folders. If you have installed the game into a non default directory this might be something to look into.

Every GOG game can be run by a shortcut to the game exe direct--by passing Galaxy completely. I use Galaxy only for installing and updating, but I run the games through a desktop shortcut direct to the game executable. Galaxy is not required to actually run any GOG game.