I have never been taught this in particular, but it seems unlikely that the Pauli exclusion principle could do it. It's a symmetry, not a force.
From what I understand, if you sent two fermions at each other, assuming they don't otherwise repel, they'd just pass through each other. The Pauli principle would merely guarantee that they do so at an anti-node. You'd never find them at the same spot. You also wouldn't find them at any other anti-nodes that appear along their trajectories, or more accurately, their joint trajectory in configuration space, or still more accurately, their joint waveform in configuration space. In any case, their momentum and energy would be completely unaffected by this.
The Pauli principle might be why electrons end up in a pattern in which they repel each other so well, but I don't see what else it can do.
If I'm wrong, please correct me, and send me somewhere where I can read more about how it works.
I thought this video was a really good question dissolving by Richard Feynman. But it's in 240p! Nobody likes watching 240p videos. So I transcribed it. (Edit: That was in jest. The real reasons are because I thought I could get more exposure this way, and because a lot of people appreciate transcripts. Also, Paul Graham speculates that the written word is universally superior than the spoken word for the purpose of ideas.) I was going to post it as a rationality quote, but the transcript was sufficiently long that I think it warrants a discussion post instead.
Here you go: