Neil Armstrong, 1st mayor of Luna City in 1986?
There's a nugget of truth there. Without Vietnam, money which was spent on that war (which is a hell of a lot of it) could be instead turned to civic matters. Among them, space. As it was, had we continued Apollo in the OTL instead of going with the shuttle, we'd have Lunar bases and likely have been to Mars already by now. And much more than even that is feasible if the money from Vietnam instead is utilized to foster space exploration.
Ummm.... Without the war, NASA's budget might well have not been gutted, true. Without Nixon, Apollo may well have continued for a bit.
However, NASA was a victim of its own success, in many ways. Responding to Kennedy's challenge, they built the Saturn V rocket, with by FAR the largest engines ever built, with unheard of complexity, and it worked.
So, when it came time for the next step, they figured they could do a space shuttle - without doing step by step evolution - just build it.
Oh, sure, without Vietnam, the *Shuttle would be far more sensible than the current one - there'd be liquid boosters that would be reusable (although probably with more refurbishment and cost than expected), but they'd still have the ceramic tiles for heat shields that were such a glorious (in a tech dream way) answer to the heat-of-reentry problem. The shuttle is still likely side-mounted, and those tiles are still going to get slammed by chunks of ice, and we're still going to lose a couple of shuttles. The tiles are still going to be lost, and the jigsaw-puzzle shapes are still going to mean replacements are made on a one-off basis.
A fully reusable shuttle means even more complexity in the turnaround, with even larger staffing.
A reusable shuttle is still going to be built by the aerospace primes on a cost-plus basis and cost a fortune.
Also, a 'shuttle' means that if you put 100 tons in orbit, 80% is still going to be orbiter, leaving only 20% for payload.
OTOH, a heavylift cargo Shuttle-C equivalent could probably help with the last problem.
Also, the Apollo project was a huge 'get to the moon', 'beat the Russians' effort, and once the US GOT to the moon, the funding would be slashed significantly under ANY administration. OTL NASA took decades to understand that the tap wasn't going just be turned back on to full flow, and iATL, with cuts being restrictions rather than slaughter, they might take even longer to recognize reality. They will keep producing proposals that cost more (in estimate) than Congress is prepared to give them. And, of course, this being the aerospace industry, those estimated costs will be a fraction of actual costs.
Once their mission succeeds, NASA is also going to evolve from a movement into an organization. You're going to get the same bureaucratization as OTL (probably even worse, as it's bigger). You're likely to get the same denial of engineering realities that led to the Columbia disaster. Etc.
All in all, a 'no Vietnam war' scenario means a LOT for space. There's probably a real space station, or multiples in orbit. There's probably some sort of moon base, possibly intermittantly visited, with maybe MAYBE as many as a dozen astronauts. Cost to orbit is probably 1/2 - 1/4 of OTL.
But landing on Mars? Luna City? the promised 'revolution' in launch costs to LEO of 1/10 or less? Close to being ASB, I'm afraid, as much as I'd dearly like love to see such.
(Note that a bunch of studies were done in the 70s that predicted that 1/10 launch cost would be a qualitative industry changer, that whole new markets would be available at that cost, where 1/2 cost would just be more of the same.)