Unless it is a nuke there is no missile that is going to sink a CVN.
Our ASROC had nuclear capability in the very early 60’s. You think the Chinese won’t send their very best when they can take out a CVN with a nuke?
Reportedly the U.S. Navy believes the Chinese Dong Feng ballistic missile, specifically the DF21-D may have the ability to sink capital ships including carriers from a range of up to 2,000 km. The warhead comes almost straight down on its target at speeds in the neighborhood of 4,000 MPH (about twice the velocity of a bullet from a high-powered rifle).
A carrier has few defenses against an attack from such warhead in comparison to the counter-measures it can bring to bear against other incoming threats.
Chinese are now believed to have a sufficient number of Beidou satellites (COMPASS-G2) in orbit to make precision tracking and targeting possible.
This information is from 2011, so presumably the U.S. is developing the appropriate responses. The OBVIOUS response is to take out the Chinese targeting satellites.
There might also be a cyber countermeasure capable of interfering with the targeting.
BUT WAIT . . . we have a “leader” in the White House who thinks the Chinese are our friends (along with peace-loving Muslims) and the real enemies are ignorant, racist, xenophobic Americans who voted for Trump.
According to President ZERO, the only reason the Chinese would have created the carrier-killing DF21-D is out of concern for possible future American imperialist ambitions.