Christian over at Defense Tech broke the story of a Chinese anti-ship ballistic missile program. It is an intriguing thought, but this idea just brings back too many memories of reading Payne Harrison's Thunder of Erebus.
We all know that you don't have a real Anti-Ship Ballistic Missile until you mate a torpedo to the warhead!! Close-In Weapon Systems cannot yet deal with torpedoes!
But in general, yeah, current generation CIWSs are designed to deal with the terminal "diving" maneuver of modern anti-ship missiles. So they have the maneuverability to acquire and track something diving down at them from space. The current technical challenge is that of mass: Can they pump out enough bullets/fragments to shred the re-entry vehicle?
At this time, the fastest countermeasure would be to network multiple CIWS together, and target one RV at a time. The current ammo capacity on the CIWS is definitely insufficient in a saturation attack scenario.
In the future, we can expect to see more multi-gatling CIWS like the Russian Kashtan or the Italian Myriad. They have to make up for the Gatling gun's low initial firing rate by adding guns.
Update: Yes, I know you're going to bring up Metal Storm as a potential solution. No, Metal Storm is NOT a solution to this. Their design has a serious limitation to ammo capacity. Just imagine lining up 400 20mm bullets from end to end, and you can see that they are unrealistic.