No, that's not how signing works - encryption is a function of a message and a public key, such that you can take the encrypted message and a private key and recover the cleartext. Since the public key is known to everyone, an attacker can modify the message, generate their own hash, and encrypt that with the public key, just as easily.
It's true that you can construct a signing scheme by using a decryption algorithm: treat the hash as if it were a ciphertext and "decrypt" it with the private key, then anyone can verify it by "encrypting" it with the public key. That way only the person with the private key can sign it, which is the direction you want. But this doesn't run afoul of the FCC's rules, because you're not obscuring the message; everyone has the public key.
If you're using so-called "textbook RSA" (i.e., the type of RSA you can explain on a whiteboard, with no hardening against real-world attacks), it's also true that the encryption and decryption operations are the same mathematical function. But you don't actually want to use textbook RSA, at the very least because of malleability (if you multiply two textbook-RSA ciphertexts, you get the same thing as if you'd decrypted them, multiplied the cleartexts, and encrypted that, except you don't need to know the private key), but also because of determinism, timing attacks, etc. Real-world RSA signature algorithms bear little resemblance to real-world RSA encryption or decryption algorithms.