Under RSA protocol I was wondering how a message sent could be verified under the following scenario
RSA is not a protocol, it's a crypto system. It contains several algorithms that utilize the same trap door function but different encryption and signature generation methods.
Can Bob send Alice a second message in plain-text to determine what he had originally sent?
Of course. He could also send a hash of the message he send, so that an attacker cannot view the message that was send, but note that brute force attacks may be possible. In general it doesn't make sense to send a message in plaintext if you're after confidentiality.
Does Alice send back Bobs message encrypted using her public key so that bob has cipher-text he can compare? Can bob only compare cipher-text? or can he do anything else?
No, because a ciphertext is supposed to be CPA-secure. That means that - in general - a random padding or IV is used. So the ciphertext should not give any information about the plaintext, including equality of the plaintext.
With messages bob sends or receives is there anything bob can do to verify/check if the two message he sent are the same or not?
No, not unless he had access to the first plaintext message, but it seems that's not envisioned in your question.
Can Bob decrypt a message with just the public key? Is he allowed to get private key from alice?
No, and no.
This question contains answers for generic RSA. You may want to have a look at homomorphic encryption to see if anything in there suits your needs. Currently it is not clear to me if homomorphic encryption would meet your requirements.