Dof is proportional to the square of the focal length
You have to adjust focal length in proportion to crop factor to be considering the same field of view; changing it by square root of the linear scaling factor is considering DoF for a different framing of the subject.
It is time to look at some actual equations; a nice reference ishttp://dfleming.ameranet.com/
and in particular the simpler, approximate DOF formulas athttp://dfleming.ameranet.com/equations2.html
These become inaccurate only for extreme close-ups.
What they say is
a) the DoF is approximately determined by the subject distance and the hyperfocal distance H, where the latter is the distance at which to focus in order to have everything out to infinity in focus.
The nearest and farthest in-focus objects are at approximate distances S.H/(H+S) and S.H/(H-S),
S = subject distance.
c) The hyperfocal distance is in turn given exactly by
H = F^2/(N.C) [first form]
H = F.A/C [second form]
N = F/A = aperture ratio
A = aperture diameter
C = maximum allowable circle of confusion on the image formed in the camera in order for something to be considered in-focus.
So yes, when working with a fixed format, and so with fixed circle of confusion, then for a given aperture ratio and subject distance, DOF varies inversely with the square of focal length.
But when you compare different formats, it is more useful to think of it this way:
i) adjust focal length in proportion to linear format size [or in-camera image size after cropping, if any] to get the same field of view for the image.
ii) adjust allowable circle of confusion C in proportion to focal length, to get the same maximum "print CoC" size for in-focus objects, on prints in which subjects appear at the same size;
A) if you keep the same aperture ratio N, the first form shows that the hyperfocal distance increases in proportion with focal length and so with with format size: DOF decreases with increased format size at fixed f-stop, and
if you keep the same aperture diameter A (by changing aperture ratio in proportion to focal length), the second form shows that you get (approximately) the same depth of field.