The ability to appear offline on Battle.Net is a feature many users have been requesting for some time and perhaps it will eventually happen thanks to Overwatch.
In a post on the Overwatch forums, Jeff Kaplan addressed the problem which would also affect Diablo 3 players.
We would like to add an “Appear Offline” option to Overwatch. We are in the process of coordinating this with Battle.net and the other game teams. Because our social system is a cross-game system (WoW, Hearthstone, Heroes, SC2, D3) adding this functionality is not trivial for us. I know there has been discussion around this feature for a long time, and things have definitely not moved at a rate which is acceptable, I can assure you we care about the feature and want to figure out how to make it happen.