How to open/use/enable Spotify Desktop Web Player on iOS?
Try iCab http://itunes.apple.com/us/app/icab-mobile-web-browser/id308111628?mt=8

It will allow you to set the "User Agent" of the browser to pretend that you are using a different web browser, so you can tell Spotify that you are using the "desktop" version of Safari on the Mac, or Firefox on the Mac, or even Chrome, etc.

You can check your browser's User Agent at http://ua.luo.ma/ too. If it says anything about "iPhone" then Spotify will probably give you the "Unsupported Browser"