Update: There is a now a way to play full tracks! Read more at Spotify’s Developer site, and @arirawr’s answer to this question.
Previous answer: No, there’s currently no way to play the full song using the Web API. If you want full tracks to be playable from a website, you can use the Spotify Play Button. If you want to build a mobile application, you can do playback of full tracks using the Android and/or iOS SDK.