Speaking about encryption in general, not limiting it to RSA, if a man-in-the-middle modifies an encrypted message, the message will not decrypt as the same message you sent. It really depends on the encryption algorithm used. For example, if a block cipher was used, only the block with the altered data will fail to decrypt properly, the rest of the blocks would decrypt fine, though there is no way to tell which was altered. Other types of encryption would just fail to decrypt a message if a byte was altered.
That said, the primary purpose of encryption is to provide confidentiality, as in it's used to ensure the conversation between you and the other party cannot be intercepted and read by anyone else. What you seem to be asking for is a way to ensure the data you sent was unmodified in transit. Encryption can help with that, from simple basic hashing, to utilizing DSA (digital signature algorithm). If you take a hash of your message, place the hash at the end of your message, and encrypt the whole thing, when the receiver gets the message and decrypts it, they can rehash the message and verify it against the provided hash. If they are the same, they can say with near certainty that the message wasn't altered in transit.
Side note - RSA is slow, and usually reserved for tiny-message uses, such as exchanging a symmetric key, or "signing" a hash of a message. It's not normally used to encrypt large messages.