I run Overwatch via the desktop shortcut it created when I installed it (Overwatch Launcher.exe). Until relatively recently (last few weeks) I never had an issue with it.
However, since then sometimes the little popup that appears will just get stuck during loading (I think it says ‘starting Battle Net’ or something), it won’t load the Battle Net app. I can close the popup and it says ‘Cleaning up’ but doesn’t ever go away (I waited a fair amount of time).
Monitoring it in Resource Monitoring the app doesn’t seem to be doing anything on the network or the hard drive. It just uses about 20% CPU and never loads anything.
After Ending Task on it, and re using the same shortcut, battle net will usually launch straight away and I can then click Play and load Overwatch with no issues.
edit: September 14
This is still an issue for me.
It seems to mostly happen after updates to the bnet client.
Today, after I had to close the prompt, Overwatch wanted to update itself, but it said it was waiting for another installation to finish (there were no other updates for any games). I had to force it to update.