No, it cannot
#No, it cannot ThereThere is no such mechanism in RSA.
Even if you have the private key you can't check if a message was encrypted using the corresponding public key unless you know something about the plaintext (for example, that is should be readable ascii and not random bytes).
If you implement message authentication (i.e., the message is also signed using the senders private key), you can use the senders public key to verify that the decryption with the receivers private key was successful.