TL;DR Anyone can send a message and say it is from Alice if it is not signed; there is no way for Alice to prove that she didn't send those unsigned messages. However, Alice can prove she sent a message by signing it with a private key. Similarly, anyone can say Bob read a message, and Bob can't prove he didn't. But Bob can prove he read a message by signing it with his private key.
A small side point, I'm going to make a distinction here between having received and read a message. I can send an email to your mail server, (and the network will confirm that it was sent), but if your server is ignoring all incoming data, even though the network says it was received by your firewall, it doesn't mean the mail server actually read the message or has access to it in any way.
In good cryptography, it's assumed everyone can receive the message, but only the indented recipient can read it. Even though Alice here isn't actually encrypting specifically for Bob, we will assume that the message is posted in some public way that it guarantees that Bob could receive it if he wanted to, but encrypted so only Bob can read it (even though you might not technically encrypt it, just require Bob to log into Gmail with his username and password, but the same principle).
In such a case, there is no way for Bob to prove he hasn't received the message; after all, Bob could download (receive) the email from Alice from the mail server to his computer, but not actually open (read) it. If the message was encrypted, he could save the message and his private key to a non-networked computer and decrypt and read it there, burning the computer after he is done. No one could know if he successfully successfully decrypted and read the message or not.
However, at any time, Bob can prove that he has read the message by signing the decrypted message Alice sent, and sending back the signature. The only way he could have access to and sign the decrypted version (which would include having read the non-encrypted email from his Gmail account) was if he had read it.
Addendum: Ideally, Bob would sign the message with his own private key, but this isn't strictly necessary. As long as the message "encrypted" (or sent securely) in some way that only Bob can read it, the only one that could possibly have access to and sign the "decrypted" message is Bob*. If the message is not encrypted or sent securely, anyone could sign Alice's message with any key, claiming to be Bob that signed it.
* Technically, there is a second person who could sign the decrypted message: since Alice wrote the email to Bob, she has access to the unencrypted message and could sign it, claiming to be Bob. If she just wants to know if Bob read her message, this would be silly of her to do. However, if Alice wants to prove to the authorities that it was Bob that read her email, Bob would need to sign the decrypted message with his private key (which Alice would not have access to).