Theoretically, yes. But ultimately, the President has to authorize a nuclear launch, so it also depends on the President's nerve and decision-making capabilities.
If someone did have designs on launching a first strike against us, I don't think it would begin with an all-out salvo of missiles. They could send in operatives and covert teams to commit acts of sabotage of key points of communication and other strategic targets which may be vulnerable on the ground.
The typical Cold War scenario is that it would invariably start off small and then escalate from there. For example, if Soviet Bloc forces poured into West Germany, it's possible that battlefield commanders could launch nuclear artillery on the advancing Soviet troops to prevent being overwhelmed by numbers. The Soviets could then open up with their own nuclear artillery, or up the ante and start nuking NATO bases, possibly cities.
We're not really imagining a scenario where some national leader just wakes up one morning and decides, "Hey, I think I'm going to nuke the world today." (Although Reagan used to joke about that sort of thing.)
What if Putin launches a nuclear weapon on Kyiv and/or other Ukrainian cities, just to end it once and for all? What should the West do in that scenario? If no NATO cities or territories are struck and we just decide to let it pass, then our own territory would not be at risk. We could still survive, and so could the world. But if we decide to respond in kind with nuclear strikes against Moscow or other targets (in retaliation on behalf of Ukraine), then they will send nukes against us, and that will be that.