I tried to avoid it, but we seem to need formulas! Here they are, put into a form that I find more useful when comparing different formats. As noted above, I will ignore aspect ratio differences and so measure sizes entirely in terms of diagonal lengths.
The bottom line is that the aperture diameter alone determines the DoF close enough for most situations, though in practice you will often compute it from the focal length and aperture ratio, so formulas often are stated in terms of those quantities.
A) Measurements that go into DoF formulas
sc = camera format size: diagonal length of the image formed in the camera
cc = camera CoC limit: maximum allowable diameter of circle of confusion in the image in the camera in order for the subject to be considered in focus
sp = print size: diagonal length of the image on the print on which DoF is to be assessed
cp = cc*sp/sc = print CoC limit: maximum allowable diameter of circle of confusion as it appears on the printed image
f = focal length of lens
A = aperture diameter
r = f/A = aperture ratio (f-stop)
D = subject distance from camera
S = subject size: the diagonal length of rectangular region at the subject distance that ends up in the print.
When considering a print of a given size, what counts for appearing in focus is the "print CoC", so I will consider cp as fixed, so that the image CoC limit cc varies in proportion to the format size sc.
Hyperfocal distance.
The easiest case for measuring DoF is when the camera is focused at infinity; the hyperfocal distance H is the distance to the closest object that is in focus:
H = f*f/(cc*r) = A*f/cc = A * (D/S) * (sp/cp)
= aperture diameter * (subject distance/subject size) *(print size/print CoC limit)
Note: everything on the right except aperture diameter will in general stay the same if you aim for prints of a given size and sharpness of the same subject taken from the same position, so when focusing at infinity, the aperture diameter alone determines depth of field exactly.
If instread you change subject distance but keep the same framing (subject size S), hyperfocal distance H varies in proportion to D.
C) Focusing at a finite subject distance, D
When focused at subject distance D, the formulas quoted by Leon Vick from the Van Nostrand's Scientific Encyclopedia simplify to
near focus = D^2/(H+D)
far focus = D^2/(H-D)
These are actually slight approximations which break down in close focus situations but are close enough otherwise. The exact formulas, derived from
www.normankoren.com, are
near focus = D(D-f)/(H+(D-f))
far focus = D(D-f)/(H-(D-f))
which show that in close-up situations, smaller formats (smaller focal lengths) gain a bit more DoF.
What the formulas show is that under the circumstances described that give equal hyperfocal distance H, one also has almost the same range of in-focus distances when exact focus is set at the subject distance D, but with a slight increase in DoF for smaller formats, only noticable in close-up situations.
D) Main practical conclusions
When the subject distance is far less than hyperfocal, the DoF (the difference between these distances) can be described as
DoF = D^3/H^2 approximately. Thus,
I) DoF varies about inversely with the square of aperture diameter.
If you vary subject distance, D and H vary in proportion, so
II) near focus, far focus and DoF all vary in proportion to subject distance D.
III) Once you know these two numbers, D and A, almost nothing else has any significant effect on DoF, except in close-ups, where DoF is a bit greater for smaller formats.