The question of depth of field in relation to format size comes up quite a lot. The general rule is that for any given field of view and aperture, the smaller the format the greater the depth of field. This holds true for telephoto, normal, wideangle and macro work, though when you are shooting macro there are some qualifiers.
The main qualifier is that when you are shooting macro, the same field of view is NOT the same as the same magnification, a point that is often missed. Magnification is the ratio of the size of the actual physical size of the image of the object being photographed divided by the actual physical size of the object.
To make this a little clearer, lets say you are trying to take a macro shot of a postage stamp that measures 36mm x 24mm. If you use a 50mm macro lens which has a maximum magnification of 1x, with a full frame camera you can get an image that's 36mm x 24mm and that will just fill the frame. Now if you put the same lens on an APS-C camera and you adjust things so that you're just filling the frame again (which means increasing the distance between the lens and the stamp), you are now forming an image that's the same size as the APS-C sensor, i.e. 15x22.5mm. The magnification mow is 36/22.5 (or 24/15), which is 0.625x. It's not 1x like it was for the full frame case.
If you shoot at 1x with the APS-C format camera, you'll only get a 15x22.5 section of the 24x36mm stamp. You're still at 1x of course, but you see less of the subject than you did with the full frame camera because the frame is smaller.
Now the usual rule that DOF increases as format size decreases still applies, but it applies to the "same field of view" case where you're shooting at 1x with full frame but only 0.625x with APS-C. Compare the yellow and green sections of the table below. However, if you shoot for the "same magnification" case, the full frame image actually has a greater depth of field then the APS-C case. Compare the yellow and blue sections of the table below.
|Format||Area Imaged||Magnification||Aperture||DOF (mm)|
So as you can see the difference in DOF between APS-C and FF depends on how you are defining your macro work. At the same aperture and for the same field of view, APS-C has a greater DOF, while for the same magnification, FF has a greater DOF.
Note that sensor resolution doesn't really come into play here, because at the DOF limits all digital sensors have a pixel size significantly smaller than the circle of confusion value, so they will all fully resolve the detail in the image at the limits of focus. Typical COF values are 30 microns for FF and 19 microns for APS-C, while the pixel size for even an 8 MP APS-C DSLR is around 6.5 microns and that for a 12 MP full frame sensor is around 8 microns.
Note also that you can't keep on getting more DOF by just stopping down. Stopping down increases diffraction and makes the smallest possible focused spot larger. When that size is greater than the COF value (indicated by *** in the table above), it means that nothing in the image will be acceptably sharp (which is part of the definition of DOF), since even at the point of focus the sharpness will be diminished enough to cause visible loss of resolution.
So which is better, full frame or APS-C. Well, with full frame you can always back off to the point where you have the same magnification as APS-C (0.625x in the example above), then crop the image and print. You can't go the other way with APS-C. So overall you have more options with full frame (as long as you have enough pixels left after you have cropped). However from a strictly practical point of view, unless all of your photography depends on critical amounts of DOF, it really doesn't matter that much to most photographers.
If you really need more DOF than simple optics can give, the best option is focus stacking where you combine the sharpest parts of multiple images taken with different focus settings. Of course this only works for a static subject, but then most macro work is done with static subjects.