This is my answer on Quora - thought of posting it here. I just aim to put facts into perspective; would appreciate if you let me know in case my tone is not neutral at any place in the below answer.
Contrary to what many people understand, US involvement in Vietnam has been since the time WWII ended, initially as an ally to French. This initial period ended with the exit of French and division of Vietnam into communist - north, and anti-communist south.
The American involvement restarted sometime around 1965 after observing gradual infiltration of South Vietnam by the North Vietnamese army, and continued over next decade under 4 different presidents. (1965-1975)
U.S. went as far as it could - dropping more bombing material than one could comprehend and using bio-chemical warfare - however, not even a madman would have considered using Nuclear Warfare unless its sovereign interests were fundamentally threatened. Of course, U.S. soil was never threatened. However, to understand why the U.S.A. didn't nuke Vietnam, one has to truly step into the president's shoes and see how events unfolded... it wasn't all that simple.
- At no point of time, did the U.S. ever believe that it might lose in Vietnam.
- Initially, U.S. was even reluctant to use ground troops, and took either bombing mission, or very specific search and destroy missions. U.S. never believed that North Vietnam could hold for as long as it did. The key idea was that bombing and these missions could deter the North Vietnamese. After all, Vietnam was considered to be a small agri-country (which it was!)
- U.S. believed that it was almost improbable that such a small country might be able to sustain such heavy bombing and would soon back off. However, several things, such as the underground tunnels - where a large population of Vietnam lived secure from the bombings, weren't discovered till late in the war.
- U.S.A.'s primary agenda was to avoid Vietnam coming under full communist control. For this the U.S.A. was happy to have a North Vietnam and a South Vietnam. Further, in the later years of the war, as the escalation increased, both the US and North Vietnam had agreed to continue the dialogue, including a secret dialogue outside the Paris talks. However, both the sides kept on fighting in a bid to gain advantage at the negotiation table (1972 Paris Peace Accords). Nuking a country didn't fit the negotiation strategy.
- Cuban Missile Crisis (read here: Cuban Missile Crisis) and 1968 Thule Air Base B-52 crash were closely averted nuclear disasters, and had shaken the U.S. It can be adjudged that they were much more careful about using nuclear weapons and wary of starting an all-out global war.
- Further, Russia and China were both supporting North Vietnam. Though it was unlikely that Russia and China would have started a war with the U.S., one of the most powerful armies in the world, still the smallest possibility of Nuclear Warfare (something that didn't exist at the time of Japan) was enough to serve as a deterrence. Also, the overall strategy for the U.S.A was to ensure more international corporation with world's leading powers. In fact, lots of bombing campaign in Vietnam and Cambodia were conducted in secret, to avoid any escalations. Even the U.S.A.'s decision to enter Vietnam in 1965 was a secret! So the use of nuclear warheads in initial years was out of question.
- As the initial years unfolded, and it became apparent that US wasn't able to make any decisive win, the internal pressure increased. By 1969, the government did not have support of its own people on the war. During one of the protest, 4 college students were shot down: 4 million students participated in anti-war demonstrations post that in a matter of few weeks. Soon after, the My Lai Massacre spurred massive national and internal protests. Thus after 1970, the main focus was to de-escalate U.S. involvement in the war and work towards a peace treaty. Nuke attack, again was out of question.
- At the end of 1972, as the peace treaty broke down, U.S. "bombed Vietnam to middle ages", targeting densely populated civilian areas in north Vietnam. Again, the intent was to gain advantage at the negotiating table, which U.S. did. However, at the cost of huge international retaliation - Swedish PM compared the bombings to holocaust. Nuking would have resulted in U.S. losing support of its own allies.
- It's naive to think that the possibility of a nuke-strike didn't ever come up. While the previous presidents, Kennedy and Johnson, were ambivalent about how far should the U.S. get involved, Nixon (1969-1974) was initially open to use more aggressive policies. Recently, declassified documents bring to light the 'madman' theory devised by Nixon. In October, 1969, without warning, he sent B52 bombers loaded with nuclear warhead, racing to Moscow! It was a game theory move to show Soviet that the U.S.A. can go to any extent, and Soviet should back down in it's involvement in Vietnam. You can imagine, the horror of it all - one mistake to WWIII. Anyways, after 3 days of flying on the soviet borders, Nixon recalled the mission and B52s returned. Soviet neither launch a counter offence, nor reduced their help to North Vietnam. However, the point to note is that even now - it was a strategic move and not really a bid to nuke a country. Anyone who knows anything about these bombs would know that using them would have been the end of the world as they knew it.
To sum it up, while no nuclear weapons were used in the warfare, the last point shows how close it came.
Source: Wikipedia, Wired, "The Vietnam War: A concise international history by Mark Atwood."
