Actually, there the TLSNotary[1] protocol that allows you to use the https connection as a means to sign the web content your browser received. There is the PageSigner browser extension that uses TLSNotary to sign webpages.
However, it seems like this project wasn't given a lot of love this last few years. Good news is version 2.0 has been released just a week ago[2], with support for TLS 1.2, but with a major drawback for me: it now trusts a server generating the TLS keys for the notarized page. Sure, it's an "oracle" server not controlled by PageSigner but still operated by Amazon.
[1] https://tlsnotary.org/ [2] https://tlsnotary.org/wp/?p=45
A TLS session replay is unnecessary. If you've got the email with its headers intact then you can prove that you received that email.
You might want to learn what email headers are available to you. You'd be interested in learning about ARC-Seal, ARC-Message-Signature, ARC-Authentication-Results, and DKIM-Signature. Those headers will let you cryptographically validate that a message is authentic -- that the email is unaltered as Google received it from whoever sent it.
You'll also be interested in learning about Key Transparency: the signatures are created using Google's PKI which should be listed in any public key transparency log. That will let you prove that the signatures you validated were created using something that only Google knows.
Well technically whatever sent it. That stuff is all about the identity of mail servers, not individuals.
Actually, you can get DKIM pubkey from DNS, and verify the e-mail. My mail server does that upon reception, and appends an extra header with validation results to the message.
So in my e-mail client I see:
Authentication-Results: ....; dkim=pass (2048-bit key) header.d=gmail.com [email protected] header.b=sKwsgxpv
on e-mail from gmail users.
So unless Google claims I hacked the gmail or the account of the sender, I can't fake that e-mail message and make it validate.
You can subpoena Google for the records. It would probably be easier to rely on that in court than cryptographic proof, as the legal system understand that much better.
You have a green padlock in your URL but can you prove in court you received a given email? No, because a SSL connection is transient and you can't replay it to show that Google's certificate digitally signed that email in GMail.