This is not what you asked, but if you’re looking to host a static website by pushing to Git in particular, you might be interested in Netlify. They’re essentially a build step + static host. If you like Github pages, this might be up your alley.

Gitbucket - http://github.com/gitbucket/gitbucket

It has a "pages" plugin, at the time of writing it is mentioned in the README

I believe only gitlab