Is AES-CBC mode not secure against chosen cipher-text attacks even if the IV is not predictable?
No, unpredictable IV won't solve the problem (cihpertext is malleable).

One example to demonstrate the insecurity:

in the CCA game, the adversary sends $m_0,m_1$ to the challenger, and  receive back the ciphertext $c=(r,s)=(r, E_k(r\oplus m_b)$), where $r$ is the random IV. 

Then the adversary sends $c'=(r\oplus t,s)$ where $t$ is a random string.

The oracle will decrypts becausee $c'\ne c$, and the adversary now gets
$$
D_k(s)\oplus r\oplus t = D_{k}(E_k(r\oplus m_b))\oplus r\oplus t=m_b\oplus t
$$ 

The adversary then compute $m_b\oplus t \oplus t =m_b$ and can decide $b$ by comparing the result with the messages it sent.

One way to obtain CCA security is to use a message authentication code and the encrypt-then-mac procedure (authenticated encryption).