Why did America go to war with Vietnam?How did they lose? And why did they not use the nuclear bomb?

The Vietnam war is a complex topic, and I don't to get into the long road to its origins or why the US lost, but let's talk about nuclear weapons in Vietnam, which is a relatively bounded issue.

There were, throughout the war, insinuations that the US could use nuclear weapons. There were even some within the US military, at high levels, who thought they might "help." However these voices were largely disregarded by many military leaders, and certainly the important political ones. Why? For several reasons.

One is that by the 1960s, the "nuclear taboo" was well in place. Nuclear weapons were not considered "usable" military weapons. They were primarily seen as deterrent weapons, with high political and moral stakes attached to them. The US already had issues with international support of its efforts in Vietnam; had it brought nuclear weapons into the picture, it would have had to deal with very angry protests from the West Europeans, the Japanese, the Indians, and probably far more. This would not have helped it, diplomatically, from a Cold War perspective.

Would it have helped militarily? That's also not clear. Tactical nuclear weapons were developed for use against concentrated hard targets, like tank formations or major bases — the sort of thing that one might be expected to fight if one were going up against the USSR in eastern Europe. This was not the kind of enemy they were fighting in the Vietnam war. In most cases, the US won any true "conventional" engagements it fought — the problem the US had in Vietnam was not that it couldn't beat Vietnamese tanks or fighters, but that it couldn't do so and also avoid massive collateral damage, and that it couldn't win the "hearts and minds" of the Vietnamese citizens. Tactical nuclear weapons would not have helped this.

Could it have used them strategically, to cause mass slaughter in the North? Sure — and the US did engage in heavy bombing campaigns against the North, occasionally. But these came with political costs, and a risk of escalation of the war (the US was very afraid of bringing China into the war in a big way, which is why it never attempted to invade the North). And would come with all of the political condemnation already alluded to.

And — here's a real kicker — might not have ended the war. It is not clear that bombing campaigns by themselves win wars, even nuclear ones. The only example we might have of this is World War II, which was a very different sort of situation anyway, and even then it is unclear whether the atomic bombs were what caused Japan to surrender (the invasion of Manchuria by the USSR seems to have had at least as great an effect). So imagine a situation where the US uses such weapons, kills tens to hundreds of thousands of people, and the North Vietnamese go on fighting. That is a pretty ugly picture.

What other downsides might there have been? The JASON advisory group wrote a very careful analysis of the utility of nuclear weapons in Vietnam, and concluded that it not only was not at all clear that they could be decisive, but also warned that if the US "normalized" the use of tactical (much less strategic) nuclear weapons in proxy wars, this would in the long run only hurt the US. Why? Because tactical nuclear weapons, as noted, are best used against large, built-up, centralized bases and concentrations of troops and materiel... which is exactly what the US military used as its primary means of deployment. So the US in the long run benefits from the non-use of nuclear weapons far more than it loses from not using them, because the US would be the perfect target for such weapons. (The same goes for strategic weapons — the US has most of its people, and most of its capital, concentrated in dense urban areas on the coasts — extremely vulnerable to a nuclear weapon delivered by boat.)

So it was pretty clear at the time that though nuclear weapons could superficially seem like a way "out" of the "quagmire" of Vietnam, in reality they probably wouldn't have made that much of a tactical difference, and the potential negative consequences, both politically and militarily, would be huge. Though they are very powerful, the military and political utility of nuclear weapons is far more limited that might at first appear, once you start to take into account all of the potential things that can go wrong, even without the threat of escalation to strategic nuclear warfare (that is, even without mutually assured destruction), which of course always looms in the background of any imagined nuclear conflict.

On the nuclear taboo and internal discussions of whether nuclear weapons could have been used in Vietnam, see Nina Tannenwald's The Nuclear Taboo, which covers many instances (Korea, Taiwan Strait Crises, Vietnam, etc.) in which the "nuclear option" was discussed and rejected by high-level US policymakers.