Losing a US aircraft carrier would be the worst American disaster since WW2. Entire countries have been destroyed for less.
What's the thought process? China sinks the carrier, and the US backs off? The US retaliates by destroying mainland air defense networks, then degrading the Chinese nuclear deterrent by hunting down dual-use launchers like the one that just fired the ballistic missile that took out your ship?
I mean if you're making the decision to take out carriers, you're past the point of no return, or close to it. Surely nukes would be top of mind at that moment.
Not at all. If they use carriers in war, they are fair game. Why all the false equivalence? You realise US retaliating by destroying Chinese military sites is expected if a carrier is sunk right? That's fair game as well.
Attacking Chinese nuclear sites is as good as a nuclear attack because it IS the equivalent of removing a secondary strike capability so in response to attacks on Chinese nuclear sites, China can only treat as a nuclear attack and perform full scale nuclear retaliation. If China does not perform full scale nuclear retaliation, they will have no more nukes to retaliate with.
Striking carriers =/= striking civilians and cities
Sinking carriers =/= nuclear war although the reactor polluting the world's oceans will be a concern albeit separate.
China will not make moves on carriers without being in response to being attacked or extremely threatened by those carriers. If the US chooses to retaliate by attacking Chinese cities using whatever excuses they like to make, China can not retaliate against US cities without going nuclear. However the threat is there. If the US do enough damage, why would China hold back? So it's quite likely in this event, the US will restrict attacks to military sites only and suffer attacks only on its own military sites and equipment. Of course nuclear sites are different because reasons above.