Is it possible to write a program that connects to an HTTPS server and archives content, but it keeps track of the session keys and the encrypted data coming from the server, and then records all that session traffic in a file. Replaying the file will allow anyone to observe that the data truly did come from that specific server, because it's signed with the cert for that server.

In other words, is it possible to get any HTTPS website to give you what is essentially a digitally signed copy of content you want to prove originated with that site? And is it true that that digital signature is easily verified to belong to the original website?