As far as I know, with existing known technology and defenses, there's probably a number n, such that if you launch more than n simultaneous non-hypersonic cruise missiles, an aircraft carrier is going to be unable to deal with them all. This of course assumes you can launch that many missiles, in range of the carrier, simultaneously, without advanced warning.
Now, the navy also isn't dumb, so I'd venture to guess they avoid parking their carriers within that range when possible, and that they probably have some other defenses like C-130's and other ships mounted with directed energy weapons and other missile defense and whatnot. I imagine they also aggressively pursue and maintain intelligence on the location, number, and readiness state of those missiles. This starts to get complicated because there's a complicated interplay of strategy and tactics related to the off-ship capabilities, and defenses have counter-strategies (such as launching a bunch of Surface-to-air missiles at those C-130's immediately prior to launching the cruise missles) and those counter-strategies have counter-strategies, etc. etc.
I also have no idea what that number n is. It could be infeasibly high.
I would think a conservative estimate for n is roughly (number of surface-to-air missiles ready to fire in the battle group) + (number of hits needed to sink a carrier) / (probability of hitting). I suspect the first term will dominate.
If the battle group has one Gerald R. Ford class carrier, two Ticonderoga class cruisers, and three Arleigh Burke class destroyers, then it has up to 2 * 8 + 122 * 4 + 96 * 4 = 888 RIM-162 ESSM missiles and 2 * 21 = 42 RIM-116 RAM missiles, for 930 missiles total.
The carrier also has Phalanx. Three guns, 1550 rounds each, maybe ~100 rounds per kill, effective range just under 1.5 km - a distance which a missile doing Mach 0.7 will cross in under seven seconds. My vague impression is that these can't be relied on to make much difference to the numbers.
Maybe there are lasers and whatnot around now. Not enough to make a significant difference.
Note that these numbers assume that the escorts' VLSs are loaded entirely with ESSM, and that all the carrier's weapons can be brought to bear on the incoming missiles, neither of which are realistic assumptions. But it's also assuming the carrier's air wing doesn't get to do any air defence, which is probably also not realistic.
Anyway, given all those assumptions, 1000 missiles and you have a very strong chance of a kill. 2000 and the carrier is toast.
If cruise missiles cost 1.5 million dollars each (roughly what Harpoon and Tomahawk cost, although what things cost in defence budgeting is a bit of a metaphysical question), the toast option is 3 billion in ammunition, plus there will be costs for launchers and various other sundries. The carrier alone is 10+ billion, so that looks like decent value.
In reality the CSG will be unable to fire 930 missiles at a single salvo because of time and targeting constraints, as the incoming missiles will be detected simulataneously.
Also, you don't need to sink the carrier. Just prevent it from launching aircraft. Once that's done it's defenceless.
The Navy isn't dumb, and that's why they build submarines.
Carrier Groups still have an important role to play in lower-intensity conflicts, but in a Pacific war with China they'll be about as useful as Battleships were in WWII.
As far as I know, with existing known technology and defenses, there's probably a number n, such that if you launch more than n simultaneous non-hypersonic cruise missiles, an aircraft carrier is going to be unable to deal with them all. This of course assumes you can launch that many missiles, in range of the carrier, simultaneously, without advanced warning.
Now, the navy also isn't dumb, so I'd venture to guess they avoid parking their carriers within that range when possible, and that they probably have some other defenses like C-130's and other ships mounted with directed energy weapons and other missile defense and whatnot. I imagine they also aggressively pursue and maintain intelligence on the location, number, and readiness state of those missiles. This starts to get complicated because there's a complicated interplay of strategy and tactics related to the off-ship capabilities, and defenses have counter-strategies (such as launching a bunch of Surface-to-air missiles at those C-130's immediately prior to launching the cruise missles) and those counter-strategies have counter-strategies, etc. etc.
I also have no idea what that number n is. It could be infeasibly high.