All you would need for this is for someone other than JFK to get elected in 1960. Eisenhower had very carefully limited the number of American personnel in South Vietnam to what would fit into the evacuation ship that was kept in Saigon harbour fueled, provisioned and ready to go on 24 hours notice. It was clearly NOT a commitment to fight to the last man and the last round.
Kennedy came along with his Counter Insurgency fixation and "...pay any price, bare any burden..." and turned Ikes politically trivial force into 12,000 advisors, making pulling them out as the situation deteriorated politically impossible because it would be seen as abandoning a major commitment in failure. So to prevent Vietnam becoming a major American war and cultural event of the decade, you first have to have someone in the White House who is willing to order the 700 or so advisors Ike had there to go to the ship and leave because the south is not saveable without making a commitment the place isn't worth.
If you make that change then you are changing a lot more than the war. No JFK presidency, no/different Bay of Pigs, a different Cuban Missile Crisis, no "who shot Kennedy" consipracy nuts... No CIA using Air America to transport Heroin out of Laos in exchange for the local warlords fighting the Pathet Lao with all the nasty consequences flowing from that.
Weaker counter-culture/hippy/rock music rebellion without The War to fuel the rage. The Vietnam war was crucial to providing emotional fuel to a lot of 1960's youth culture around the world.
Of course it could end up like that David Drake novel where Kennedy survives only to create an equivalent mess in Lebanon...
I will answer with a single word: Bah. And that's the nicest word I can say.
Eisenhower was the President who started the American commitment in Vietnam in the first place with those advisers. And Ike told Kennedy to make the Communist threat in South East Asia a priority. Kennedy increased advisers, but the conflict still remained a non-issue, off the radar of public focus, a minor issue in American governmental focus, and not an Americanized conflict. And sweet Jebus, do not take that "bear any burden" quote out of context like everyone does to make it so JFK could support everything militarily up to and including WW3 because it shows evident lack of understanding of the man and his policies when pulled out of context for that purpose.
Pulling them out was in no way, shape, or form impossible, because they were only advisers, Vietnam was not an American conflict, and the public did not pay attention to Vietnam because Vietnam was not that huge an issue. Only something like 30-some percent of Americans paid any attention to Vietnam, and of that a majority believed it would either end in stalemate or the quick fall of Saigon. It was a non-issue and not, as you said, a major commitment. Kennedy did not want to see Vietnam fall, neither as Eisenhower did, but he did not want it to become an American war or a major American commitment where the US was forced to take total care of the South Vietnamese. Kennedy being fearful of that is the reason he kept Vietnam out of the public focus so that he could more freely deal with the conflict without being pressured to increase commitment, and why he wanted to withdraw those advisers in phased shifts after 1964 and by 1965. McNamara didn't think that feasible, and was working on a proposal for withdrawal in full by 1968. And it would not have been impossible for aforementioned reasons. And even after Diem's assassination, because Diem was not a great leader and Vietnam was not functioning greatly. Diem was just the best turd, but Vietnam was then and after still a heap of sh*t on a giant platter. And there were also ideas for neutralization of Vietnam in exchange for neutralization of Cuba between the US and Soviets which could help.
To prevent Vietnam from becoming an American conflict, you need a person as President who will not make it an American conflict. Johnson was not that person, because he was ignorant of foreign policy. And he, not Kennedy, not Eisenhower, not Truman (where this whole saga started) was to blame for it becoming an American war because under no one but Johnson was it made into an American war.
Now what if Vietnam remained a proxy war between a Soviet and Warsaw Pact backed North Vietnam and a US and NATO backed South Vietnam? Would the South Vietnamese have stood a chance? Would the North Vietnamese be able to conquer South Vietnam?
Well I have two thoughts. The first is that without the United States the South would actually be forced to rely on itself more, and thus function by itself better. But, with American supplies, it would still have strength. And thus it would do better or the same against North Vietnam. The second idea is that Vietnam was not going to get over corruption, that corruption and mismanagement would eat away at its ability to fight, and that South Vietnam would fall to the Communists.
There's also the chance of a coalition government if the South Vietnamese are willing to join with North Vietnam in a peaceful manner, while being willing to lose American support.
And what about America? With no fiasco in Vietnam I can see culture being far different.
No Vietnam means less militancy, less distrust of the government, none of that polarization and conflict and vitriol between those who support it and those who don't, and a lot of that ire in the decades of the war going bye-bye. But, keep in mind you still have all the other social issues back at home. You'll still have hippies and protests and old generation vs. new, and the New Left, but it'll be more flower power. Overall, the youth will still rebel, and social issues will still be addressed and still be fought for and those will still be a source for a lot of developments and fighting as much as they ever were.