After going through the docs for Spotify Web APIs which is launched today, there doesn't seem to be a way for full length song playbacks, even after authorization. The API seems to only support 30 second song previews.
Is there a Spotify SDK for javascript to enable full length song playback on web browsers?
In a word, no. Only the native Spotify APIs (iOS SDK, libspotify etc) offer full length playback. The Play Button allows you to embed a little player on in your website, but it's essentially a remote control for the Spotify Desktop Client or the Web Player.