Does your game connect to a server? Do you have push notifications in your game? Do you have every user's email address?
If you did it could be possible to update the game or at least send a message with a link to download and install. You would need to have this functionality already built into the game though and the user would have to have "Unknown Sources/Allow installation of apps from sources other than the Play Store" enabled in their settings.
Otherwise, there is no way to update your game except through the Play Store updating process. Basically, you're looking at a lot more work than if you just went through the Play Store.