Of course not. If the States will dare to use nukes against Russia, China or even India, they will be burned out in about 30 minutes. No one will survive. Even in bunkers. So I don't think that a tactical nuke attack on the US aircraft carrier group will trigger a mutual ICBM strike, because it will mean a
total annihilation of the United States and all NATO countries.
Also even non-nuke cruise missiles are able to sink a Nimitz-class aircraft carrier just easily. This is how old good non-nuclear SS-N-22 Sunburns made in 1982 work:
The modern Russian missiles like SS-N-26 Strobile, SS-N-27 Sizzler and SSC-6 Sennight let zero chances to survive to any type of the US aircraft carriers. Just compare a price of a cruise missile and a price of a whole aircraft carrier with 3000 men aboard. Now tell me you didn't know that
Once again: your aircraft carriers are cool, but they are dinosaurs in 21st century. Course they could work fine against stone-age countries like Somalia or Yemen, but not against Russia, China or India. They will be sunk in minutes with a
minimum effort. All the mentioned countries have a lot of cheap-ass shipwreck cruise missiles that are undetectable and uninterceptable, because they fly on almost a zero attitude with an ultrasonic speed. Boom and there is no "Abraham Lincoln" with all its crew. These missiles are very cheap and extremely reliable nowadays. Sad but true.