I have been trying to use the Google Drive API to make a Fusiontable publicly readable, and have not been able to get it to work. I am able to use the OAuth 2.0 Playground
If you'd be OK with a temporary workaround, inserting Fusion Table public share permission still works with older XML-based GData API. You can check it out here, though beware of the red banner at the top of the page saying 'The deprecation period for Version 3 of the Google Documents List API is nearly at an end. On April 20, 2015, we will discontinue service for this API.'
So if you need to workaround the problem now, that would keep you going till April and then let's hope the Drive API bug gets fixed before that...