OpenVZ is almost identical to LXC, which is what Docker uses under the hood. You could port Docker to OpenVZ very easily. In fact, there is a plugin planned to do exactly that.

Fun fact: before running on lxc, dotCloud (where Docker originated from) was initially based on OpenVZ (which was great). As the mainline kernel's support for containers improved (in large part, I believe, thanks to the pioneering work of OpenVZ), eventually we moved over to that.