I've never used pitch correction (yet) but it also strikes me that if you did indeed know where tones and overtones laid on the spectrum it wouldn't be that difficult to isolate the third or whatever and adjust it a semitone to where it needs to be. In fact I think I've seen this done in one of the plethora of tutorials I've watched.
Perhaps going in reverse (kind of) of what I contemplated here would be better. Find a chord that has similar properties to the one you want to achieve (in this case lets do G to Gmin). Use eqing to remove the major third and as many of the thirds overtones as possible. Then copy the track and reverse the EQ so only the removed tones/overtones are audible then use pitch correction to bring them down a semitone to achieve the minor third. Blend both tracks and the now lowered third should create Gmin.