I'm sorry Roy but that's not accurate. Go to any text on DoF or any online DoF calculator and you'll see that if you keep every other parameter the same and only change sensor size (or crop area) the DoF definitely changes. And yes it's because the circles of confusion grow in size with the image resizing.
Basically the slightly out of focus points of light that render as circles of confusion grow substantially with the resizing step and the resulting DoF decreases. And by definition the DoF for a constant output image size is defined as the points where CoCs grow to a predefined size (e.g. 0.01") for a defined image size at a defined viewing distance.
Here's a comparison of DoF between a full frame and crop body camera with all other parameters including: focal length, aperture and distance to focal plane kept constant:
Cropping or shooting with a crop body sensor and resizing for comparison at constant output size definitely changes DoF and when all else is kept constant the DoF decreases in response to the CoCs increasing in size in the enlarging step. As posted at the beginning of this thread it gets more complicated when you talk about changing lens focal length to hold the field of view constant but still the process of cropping impacts DoF.