"At the time scientists still believed there might be
   microbial life on the moon and Sagan had suggested a 
   nuclear explosion might be used to detect organisms."

That seems a very odd thing for a physicist to say, given that of course a mushroom cloud wouldn't form in a vacuum. Perhaps Dr. Reiffel was not at his best on the day of the interview?

- ICBMs can't reach the moon, or even anything remotely close to it

- Mushroom clouds don't form in a vacuum.

- The fact that some Air Force officer thought it would be a neat idea (if any of this is factual) does not in any way imply that it was even remotely considered by anyone with the authority to make such a decision, any more than seeing a person who works at the Apple store drinking a coke implies that Apple Inc is buying the entire CocaCola corporation.

  - Does the RAF [Royal Air Force] agree on canceling Trident?
  - You could ask them. If you're interested in the opinion
    of garage mechanics. They'd want to keep it. They want to drop
    the bomb from an aeroplane. They just like dropping things on
    people, but they're no good at it. Couldn't even close the
    runway at Port Stanley. Probably never find Moscow, and if
    they did they'd miss it.

https://en.wikipedia.org/wiki/Operation_Black_Buck

> World peace cannot be achieved by sitting around on our duffs singing hippy songs to the moon. Peace can only be achieved through excessive acts of seemingly mindless violence.

> [...] And all the other countries would exclaim, "Holy @$#%! They are nuking the moon! America has gone insane! I better go eat at McDonald's before they think I don't like them."

[0] http://www.imao.us/docs/NukeTheMoon.htm

Was the US Air Force capable of launching a missle to the moon in the late '50s, early '60's?

Lets say they chose the Atlas. Lets be generous and say the Atlas E was chosen (Lockheed , first launched in Feb 24, '61) [0] it was capable of an 820Kg payload into low orbit. The interesting bit, for the total life of this particular rocket there were "9 successful launches, 7 partial successful launches and 2 failures."

Hardly a reasonable plan to get a bomb to the moon. Nothing really happened until Wernher von Braun was put in charge of the F1 and by then the target was a manned mission.

[0] http://www.designation-systems.net/dusrm/app3/b-2.html

[1] https://en.wikipedia.org/wiki/Wernher_von_Braun#NASA_career

So the thought process was literally, "I wonder if we could see a nuclear explosion on the moon?"

If so, you're ignoring science just as much as the religious zealots.

Nuking the Moon meant that they could put a nuke to reach Russia and beyond.

Remember rocket science is the same if you are putting people on the Moon or just using missiles with nuclear payloads to hit an enemy nation.

It was a show of superior technology, and if they nuked the Moon it would have been one more step for longer range nuclear missiles.

Be glad that the cold war was fought in space via the space race instead of sending missiles to different places in the world or even to the Moon. Technology from the space race really innovated things and brought about the microchip and personal computers.

https://en.wikipedia.org/wiki/Starfish_Prime

We already nuked space, just for fun.

So the bomb is over 10^12 times weaker than the moon, affecting its orbit and the tides by one part in a trillion. Quite unnoticeable. Source: Wolfram Alpha

Source: Space 1999

Although, I suppose this also happens if a meteor hits the moon.

I can't imagine though that you'd get enough of it into an orbit that intersected the Earth's atmosphere to do any sort of damage. You might have some great meteor showers as it entered the atmosphere but probably no chunks big enough to reach the ground.