In baseball, has anyone pitched a one-pitch inning?

No. It is not possible to pitch a complete inning with less than 3 pitches. For a pitcher to get 3 batters out with the minimum pitches per inning, at least 3 pitches must be thrown to each of three batters, each pitch must be hit, and each hit must result in an out. If a pitcher comes in and get the third out in one pitch, this only counts as .1 of an inning, not as a full inning.

http://www.baseball-almanac.com/feats/3_pitch_inning.shtml

The only possibility of a pitcher getting all three outs in an inning on one pitch is to be a relief pitcher coming into the game with at least 2 men on base and then getting the batter to hit into a triple play.

This feat has actually been accomplished 8 times in major league history. The reference tells the story of one of those times.

http://baseballhistorian.blogspot.com/2011/06/life-and-times-of-relief-pitcher-steve.html

The reference posted above doesn't mention if the triple play of Kealey's occurred on the first pitch, but at least two pitchers -- Jeff Nelson and Ken Ash -- have done it on the first pitch:
https://www.reddit.com/r/baseball/comments/1md0lv/has_a_pitcher_ever_gotten_3_outs_on_one_pitch/

Also a relief pitcher could come in with men on base and either pick one runner off and throw a double play ball on the first pitch, or pcik two runners off and then record a single out on the first pitch.

It's also theoretically possible for a pitcher to have a zero pitch inning, if 3 runners were picked off, but I don't believe that's happened.

Actually, it'd be possible for a pitcher to start an inning, throw no pitches, and record three outs. If
a batter refuses to enter the batter's box in a timely fashion as directed by the umpire, the batter is charged with a strike. 9 instances later, the three men are out.
Therefore it'd be possible to pitch a perfect game if that scenario was taken to the absurd.
Rule 6.06:
http://mlb.mlb.com/mlb/downloads/y2012/Official_Baseball_Rules.pdf

The previous response is correct, in that a minimum of three pitches must be thrown in an inning for it to be complete - however - it is possible for a pitcher to be CREDITED with one inning of work by throwing only one pitch. It can only be done when that pitcher comes into the game in relief, with at least two men on base and no outs. If such pitcher throws the first pitch that is then hit into a triple play, all three outs would be credited to that pitcher - thus giving them one inning of work having thrown only one pitch.

I found one instance where this happened on July 13, 1995. Seattle Mariners pitcher Jeff Nelson entered the game in the top of the 9th inning after the previous pitcher gave up two hits to the first two batters. His first pitch was then hit into a triple play. See the boxscore here: http://www.baseball-reference.com/boxes/SEA/SEA199507130.shtml

Here's another time it happened - May 30, 2008, top of the 8th inning in this game: http://www.baseball-reference.com/boxes/SFN/SFN200805300.shtml

I also believe it would be possible for a pitcher to get credited with an inning of work by throwing ZERO pitches. That would require coming in as a reliever with the bases loaded and no outs, then proceeding to pick off all three runners without throwing a pitch.

I have yet to come up with any information showing where the pickoff situation has ever actually happened.