Because I also cannot connect to D3 (the launcher just sits there for a minute or two and then says "Failed to download. If problem persists contact customer support.") I figured I would ask here too since I'm pretty sure if I can fix the connection to one game, it would fix the other.
As you can see from the linked thread, I have tried resetting both the router and the modem, uninsalling AVG, turning off firewall, turning off windows defender, closing programs, etc.
The only new thing that I have tried was to delete the battle.net folder as per another support thread. That also didn't seem to work.
So if anyone has a fresh perspective on my issue and can shed some light on it, I would be most greatful!
EDIT: Oh yeah, and I flushed the DNS and opened up the wow ports on my router.