Okay just finished installing the Android APP to phone as the old Authenticator is no longer supported.
When I opened the Battle.net APP it was asking for a password. After inputting the password the screen flashed and opened another page requesting password, so basically seemed to be stuck in a loop.
The FIX is check which browser on your phone is the default and change it.
For me I was using AVAST ONE BROWSER , changing back to Chrome fixed the issue.
I had exactly the same problem, but I had the Firefox Browser in the settings.
It would be nice to have a hint or something like that which informs you, that the used browser isn’t supported. It took me quite a while to find that out and of course I found this post AFTER I found the workaround myself 
I also had this same issue. I use Microsoft Edge browser on my main PC so I also installed in on my Android phone which unbeknownst to me set it as my default browser. I decided I didn’t like Edge on my Android phone and went back to using Chrome and didn’t think to change the default browser back to Chrome. Changing my phone’s default browser back to Chrome resolved this issue- thanks BlindOldMan.