I know people fear China but they are no cold-war Russia. Anything conventional and the US is in the lead by a lot. There's no reason to discuss anything nuclear because everyone loses in that case.
There have been reports that the Chinese have developed and brought to initial operational status the DF-21D, a medium-range, hypersonic, anti-ship ballistic missile specifically designed to be a carrier-killer [1]. The concern is that our carriers might be chased from the Western Pacific.
[1] http://en.wikipedia.org/wiki/DF-21#DF-21D_.28CSS-5_Mod-4.29_... -- also http://thediplomat.com/flashpoints-blog/2013/06/12/why-china...
It's one of the reason that warships in general (including carriers) are designed to be compartmentalized, to have redundant backups (and backups to the backups), etc., is because it is expected they might have to survive battle damage.
Obviously an ASBM may change the calculus of whether you send carriers in early in the conflict (preferring instead to use submarines with cruise missiles to take out Dong Feng launch sites or whatever you'd have been trying to bomb in the first place). But I don't see how ASBM is a guaranteed single-shot carrier kill by any stretch either.
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...
How is a carrier going to protect itself from coordinated submarine attacks?
