Hm, reading up a bit about SSL/TLS, it seems this will not work. Apparently it uses a Message Authentication Code to ensure authenticity, not a signature. But that means that even if you have all the data sent by the server, you still can't prove to a third party that you didn't modify it.

I'm not sure, but possibly you could prove that you did an SSL handshake with the server at a given time, so you can prove that the server sent "something", but not necessarily the particular thing you have. :)