My head hurts reading the replies, but here's what I know from experience, and I'm avoiding the DOF Calc.
If you use the same exact lens, and you shoot the subject at the same exact distance, DOF should be the same, but the APS-C will give a tighter crop. It's the same as if you shot it with the FF camera, and then just cropped in LR. DOF does not change just because you cropped the picture.
If you move positions (i.e. getting closer with the FF or moving further away with the crop) to get similar framing, then the DOF should be less on the FF, because you are now closer to the subject. The focal length and the physical aperture size has not changed. If, however, you grabbed a 75mm lens to shoot with the FF and a 50mm lens on the crop (assuming a 1.5 crop), and you kept distance to subject the same, then DOF will be driven by the physical aperture size, so if f/stops were the same (e.g. both 1.4), then the FF should have shallower DOF because 50/1.4=35.7mm vs. 75/1.4=53mm.
That's the way I understand it, but actually don't really care, lol. I have the lenses I have (50/1.4 and 85/1.8 on FF and 22/2 on my EOS-M) and I like the look of these lenses a lot, so if I'm wrong, I'm wrong
Also, if "viewing distance" to the picture comes into play here, well then I'll just keep quiet. My viewing distance is just about always 2 feet (distnance from my chair to my monitor)