The threat is the Chinese Dongfeng-21D and newer Dongfeng-26 anti-ship ballistic missiles. The range of the DF-21D is limited (800 nautical miles) but the DF-26's range is three times that. The missiles are dual-capable -- nuclear or conventional -- but I think we can discount the nuclear threat; that would lead to a substantial or full nuclear exchange. The targeting is the rub, but according to Wikipedia, the Chinese are working hard on that aspect.
https://en.wikipedia.org/wiki/DF-21
and
https://en.wikipedia.org/wiki/DF-26
But the U.S. Navy is working on a counter: The Standard Extended Range Active Missile or SM-6. Also being bought by the Japanese and Australian navies.
https://en.wikipedia.org/wiki/RIM-174_Standard_ERAM