How do I get a block and its content commited to a Git repo?
Turn it into a Feature. Bad thing is, it does not work on built-in custom blocks, so you need to use Boxes to get it work.

That way, you can export block as a module, and modules are text files, so GIT accepts them with it's full support.

Then, there is Features to Git module. It claims:


  Features to Git allows trusted users to write features directly to disk, commit
  them to git and push them to a remote. This makes it possible for site
  builders to be able to manage features in git without having to get their hands
  dirty on the command line.