For a carrier, not-sinking != operational. The flight deck, catapults, and arresting gear are necessarily exposed to anything that explodes above them. Take them out and the carrier is basically useless.
A carrier can be severely damaged or even sunk by just a few comparatively-small bombs if they're well-placed, intentionally or fortuitously. This was vividly demonstrated at the Battle of Midway in June 1942 [1], where four Japanese carriers and one American carrier were sunk. (It only took one U.S. bomb to sink the unfortunate Akagi.)
In January 1969 my former ship, the USS Enterprise, was taken out of action for several weeks by an accidental fire on the flight deck that cooked off ordinance, spread, and killed 27 sailors [2]. (It was before my time there.)
I would imagine that a ground-launched ballistic missile could carry a much bigger payload than I've been talking about. But yes, it'd be no small navigational challenge to make sure an ASBM actually got to the (moving) target and past the last-ditch defenses.
[1] http://en.wikipedia.org/wiki/Battle_of_Midway#Attacks_on_the...
[2] http://en.wikipedia.org/wiki/USS_Enterprise_(CVN-65)#Vietnam...