you could certainly create your own script that does
git add *.* git commit "backup" yourname youremail git pushand put that in your stash/bin, which lets you do a one liner. You could also create a wrench icon that launches stash and executes a script.
It is also possible, but i really don't reccomend it, to hook into file saves, then do something there.... but trying to push to git inside such a callback probably is not what you would want.
A lot of people like the working copy app, which is git based, and supports drag/drop, and other methods of syncing.