A browser feature would be required that could calculate/display the hash of the delivered code and optionally verify it against a 3rd party server. Ideally you'd want have particular versions signed as "audited" etc.
You're neglecting the server-side code. If you have access to the full source code to verify it, you're not describing a web service, you're describing a local application that happens to be implemented in a browser.
You already can distribute signed browser add-ons.
I don't see how it's a theoretical impossibility.