The entire official nuclear policy of every single nuclear power (as far as I'm aware) is that they will only use nuclear weapons in response to a nuclear attack or an existential threat. Officially. I believe Putin stated at some point that he would not consider the use of small tactical nukes to be nuclear escalation (if Russia does it), but I would expect the US to be better than that.
US carriers carry over 5,000 sailors, having one taken out would be almost unimaginable (a one time event with more US deaths than the Iraq and Afghanistan wars combined).
The US has never had such an official doctrine, nor to my knowledge has any of the other major nuclear powers.
Really? Should the US expect it too when it regularly does the same to third countries?
Initiating a nuclear war after a nation state starts taking out your aircraft carriers is not entirely unexpected. Nuclear or not, the response would be amazingly serious.