Can I use Fabric to automatically deploy an app on my server every time I push the code to GitHub? (GitHub has the ability to POST to a URL every time I push.)
If so, how?
Sure. It's not a real good idea, but it's possible. Just setup something on a URL you control that when it receives the POST from Github, it runs your fabric script. You could use any sort of CGI/FCGI/WSGI/whatever to do this; even a couple of lines of PHP would be enough to get things underway.