I got a great response from Plesk Support. It turns out that the /etc/network/interfaces file's entry for IPv6 was incompatible. Two ways to fix it:
1) Make sure that IPv6 entry is a static address.
2) Delete that IPv6 entry and let Plesk assign an IPv6 address to that interface. With Vultr...
I'm using Vultr's one-click Plesk installation (gets me a discount). My test site only responds on its IPv4 address.
My Ubuntu 16.04 server has IPv6 configured.
Plesk dashboard is responding to IPv6
If I disable the NGINX reverse proxy and go straight to Apache, it responds to IPv6
ifconfig...