Whenever the Cuban Missile Crisis is discussed, it is in the context of the day we narrowly averted a nuclear war. Indeed, a Soviet submarine captain reported tried to fire a nuclear torpedo after depth charges were dropped on his submarine. It was undeniably a very dangerous situation. However, is it the consensus of historians that the war between the U.S. and the Soviet Union would definitely have escalated to nuclear war if hostilities broke out, or is it possible that the countries might have had enough restraint to fight a conventional war?
I mean, clearly it didn't lead to nuclear war, so it didn't necessarily have to lead to it, thank goodness.
But if you're asking, would the use of tactical nuclear weapons — either the B-59 submarine incident, or many other scenarios that could have come to pass — necessarily have escalated to a full nuclear exchange, the answer is: of course, we don't know. But everyone who was involved feared that kind of thing, because the escalation potential was huge. It's not impossible to back down from escalation (indeed, the crisis showed that), but the further you get along, the harder it gets. The biggest fears of those in both the US and Soviet high command was that some little mistake would cause an all-out shooting war to begin, or make the other side think it was about to begin, and in that situation the possibility of thinking you can blunt the other sides' attack by going first becomes very tempting.
We might also ask what "conventional war" would have likely looked like if it went up another notch on the escalation latter. We know the answer to this, because we have the US planning discussions: a land-invasion of Cuba by the USA. We can then ask what would the response to that have been, and we know the answer to this as well, because we have the Soviet and Cuban side of this story as well: the Soviet and Cuba forces on the island would have used some of the hundred-or-so of tactical nuclear weapons already on the island (that the Americans were ignorant about) to repel the invasion.
And what then? Would the US, having lost thousands of American troops to nuclear arms back off and lick its wounds? Or would it double-down? Again, we don't know. But you can see the awful possibilities here. Especially when you factor in other possibilities: the Soviet forces in Eastern Europe being moved forward to take Berlin in retaliation for US action in Cuba; US and Soviet military forces clashing without direct communication to their leaders; "pre-delegated" nuclear weapon orders, in which commanders at times had the ability to use their nukes without getting explicit permission; the fact that none of these nukes in those days had anything like physical locks on them, and could be used by anyone who had physical control over them; etc.
One can of course make up an elaborate, imaginative way to believe they would all stay restrained and polite and rational and all that. It is telling that the leaders in place at the time did not for a minute believe that was the most likely outcome.