IIRC discussion of AShBM (Anti-ship Ballistic Missiles) has been done elsewhere here on DT. Usually involving the PRC infact.
The general consensus that I recall was that such a weapon might (yes, MIGHT) be a threat, but such a weapon runs into some fairly significant technological and infrastructure-based limitations.
In the case of a AShBM being employed against a CV and/or CBG (the same thing, really) such an attack needs to be conducted OTH, against a mobile platform(s). This means that the attacking country (PRC in this case) would need to be able to get OTH target-quality data, and then relay the needed information and updates to the missile or warhead.
Given that a USN CBG has itself an organic OTH sensor footprint, then attempts to use an MPA equipped with long-range sea search radar could be completely shutdown by a CAP. If a sub was lurking nearby to relay the data, that could also be intercepted/negated. Either by reception of the comms broadcast, by the CBG ASW assets, or the 1-2 escorting SSN's which usually operate with or near a CBG. This then leaves reliance on either something like OTHR based in the mainland, or some form of satelite-based detection and tracking system. Of the OTHR systems I am aware of, while they are capable of providing a great deal of information, and make excellent 'tripwire' detection systems, they are not (currently) capable of providing target quality information. For a satelite-based system, I am again unaware of a satelite system that is currently capable of providing target quality data. There are indeed satelites which can detect an object at a location with the needed degree of accuracy, but they are not capable of doing so in 'real-time'. The US is only just now starting to deploy systems which might enable satelites to either provide or relay target quality data in real-time.
The above paragraph just covers some of the issues which would be involved in detecting, tracking and targeting a CBG. There an additional set of issues which exist with using a ballistic missile against a moving target. Amongst these issues, there is the question of whether or not a ballistic missile or its warhead could maneuver sufficiently to compensate for target movement. There is also the question of how the warhead could get the targeting updates once it has launched. This is of course assuming that the required sensor/datalink system are in place to provided the updates as needed.
In the end, it is possible that such a weapon system might be viable, but there are significant hurdles which need to be overcome first.
-Cheers