I’m having this issue as well as a Verizon user outside of DC in Maryland. I was logged out shortly after 1pm EST, and have not been able to reconnect since.
I can access these forums, but not blizzard dot com. However, I can access Blizzard’s site on my phone if it’s using AT&T’s 5G connection instead of the Verizon wifi. So at least for the main site and presumably for BattleNet, the issue does appear to be network-dependent.
ETA: Magically started working at 3:49 EST! I did a power cycle earlier which seemed to have no effect, but it is working now.