Github could require a particular license for pubic repositories if they wanted. I actually think it might even be a good idea. Not a specific license, but requiring that public repo's must specify one of a particular set of "open" licenses.
They would probably have to implement this by requiring that a license be specified for any newly created public repository, and by giving existing public repositories a deadline by which they would have to specify a license or be "disabled". Disabled as in no one can push/pull or fork the repository.
The hardest part I think is going to be existing forks of disabled repositories. Actually even going forward I'm not sure what that means. Does the "owner" of a fork get to change the license on github? What does that mean? Damn, copyright is a godawful mess and I would like it to be very different, but at the very least I'd like copyright to only apply to registered items like it did before the 1986 Berne Convention (see the wikipedia article) and require a copyright notice.