CCA-attack is possible in RSA, but how the decryption key $d$ is known to anyone CCA-attack is possible in text-book RSA, but how the decryption key $d$ is known to anyone, as it is kept secret with receiver.
Chosen Ciphertext Attacks against RSA (such as Bleichenbacher's Attack on PKCS #1 encryption) do not actually reveal the private key.

If they don't, they why do we say that they are a threat?  Well, it's because that the keys are generally not of interest to the attacker.  Instead, given an encrypted message, the attacker actually wants to know what that message says. He's interested in the key only because the keys would reveal the message.  If the attacker can find a way to get the message without the key, he has won the game.  Bleichenbacher gives him such a way.