I guess we can all agree that a screenshot of a website proves nothing; one can edit the HTML as much as one likes, take a screenshot and show a website displaying false information.
As far as I know, the way HTTPS works is it uses asymmetric cryptography in order to exchange symmetric keys, for performance reasons. This means that such symmetric key is signed by the host of the website and sent to the user, but the host isn't signing every single piece of content it serves.
The question then is, is there any way one can prove some content came from a website?