Not much; signing the GCM tag using RSA is inadequate.
For a start, knowing the AES secret key, it's easy to make a message with any desired content (except in the very end), and an AES-GCM cryptogram that checks, and has any desired tag. Thus any holder of the AES secret key can forge a message that pass verification under the RSA public key of another person, after intercepting a ciphertext signed by that person.
Also, one holding the AES secret key can prepare two very different messages and their AES-GCM encryption sharing the same tag, allowing to spread FUD by showing that the signature of one message also applies to another message.
Further, if RSA-signing the tag was done nevertheless, that must not be with textbook RSA, which is insecure when directly signing small values $m$ such as the 128-bit tag (e.g. when $e$ is so small that $m^e<n\,$, or under the Desmedt-Odlyzko attack for any $e$).
An appropriate way is sign-then-encrypt: signing the plaintext using RSA (e.g. per RSASSA-PSS or RSASSA-PKCS1-v1_5), appending the signature, then using AES-GCM on the whole. This way only the holder of the RSA private key (rather than any holder of the AES-GCM secret key) can make a valid message that pass verification under the corresponding public key; and the signature (which discloses information about what's signed) is not accessible before decryption.
Here I don't recommend encrypt-then-sign, because I do not rule out that the integrity could be vulnerable to a replacement of the original AES secret key.