Yes, a sensor's crop factor can be used when calculating the change in depth of field (DoF) of a lens compared to that lens' use on a full frame (FF) camera. But it will not always lead to an increase in the DoF. If shot from the same distance and displayed to the same size, the DoF for the crop body camera will be reduced (because the virtual image projected on the sensor, including the circles of confusion, will be enlarged to a greater degree). If, on the other hand, you adjust your shooting distance to frame the subject similarly the DoF will increase.
There are so many variables to deal with in this question and most of the answers assume several without specifying those assumptions. This leads to gross misunderstandings about the relationship of focal length, aperture, sensor size, shooting distance, display size, viewing distance, and even the visual acuity of the viewer to Depth of Field (DoF). All of these factors combined will determine the Depth of Field of an image. This is because DoF is a perception of what range of distances from the focal plane are in focus. Only one distance from the focal plane is actually in focus such that a point light source will theoretically produce a point of light on the focal plane. Point light sources at all other distances produce a blur circle that varies in size based on their proportional distance to the focal plane as compared to the focus distance. DoF is defined as the range between the near and far distance from the focal plane that the blur circle is still perceived as a point by the viewer of an image.
We ask questions such as, "How does depth of field change when using the same lens on a camera with a different sized sensor?" The correct answer is, "It depends." It depends on whether you shoot from the same distance (and thus change the framing of the subject) or shoot from a difference distance to approximate the same framing of the subject. It depends on whether the display size of the image is the same or the display size of the image is changed by the same proportion as the different sensor sizes. It depends on what changes and what stays the same in regard to all of the factors cited above.
If the same focal length is used at the same subject distance with the same aperture using the same sensor size with the same pixel density and printed at the same resolution on the same size paper and viewed by persons with the same visual acuity then the DoF of the two images will be the same. If any one of these variables change without a corresponding change to the others, the DoF will also be changed.
For the rest of this answer we will assume the image viewing distance and the visual acuity of the viewer are constant. We will also assume that apertures are large enough that diffraction does not come into play. And we will assume any printing is done on the same printer at the same number of dpi but not necessarily the same ppi and not necessarily on the same size paper.
For the sake of simplicity, let's consider a couple of theoretical cameras. One has a 36mm X 24mm sensor with a resolution of 3600 X 2400 pixels. This would be an 8.6MP full frame (FF) sensor. Our other camera has a 24mm X 16mm sensor with a resolution of 2400 X 1600 pixels. This would be a 3.8MP 1.5x crop body (CB). Both cameras have the same pixel size and pixel pitch. Both cameras have the same design and sensitivity at the pixel level. In other words the center 24mm X 16mm of the larger FF sensor is identical to the smaller CB sensor.
If you attach the same 50mm lens to both cameras and take a photo of the same subject from the same distance at f/2 (assuming all other settings are the same) and crop the FF sensor image to 2400 X 1600 pixels and print both images on 6" X 4" paper, the two images will be virtually identical, and the DoF will be the same in both photos.
If you attach the same 50mm lens to both cameras and take a photo of the same subject from the same distance at f/2 (assuming all other settings are the same) and print all of both images on 6" X 4" paper there will be some noticeable differences. The image from the FF camera will have a wider field of view (FoV), the subject will be smaller and the DoF will be greater than the image from the CB camera. This is because the FF image was printed at 600 ppi and the CB image was printed at 400 ppi. By enlarging each pixel from the CB camera by 50%, we also enlarged the size of each blur circle by the same amount. This means that the largest blur circle projected on the CB sensor that will be perceived as a point is 33% smaller (the reciprocal of 3/2 is 2/3) than on the FF sensor. If we had printed the FF image on 9" X 6" paper and the CB image on 6" X 4" paper the DoF would have been the same (both printed at 400 ppi), as would the subject sizes in both prints. If we then trimmed the center of the 9" X 6" print to a 6" X 4" print we would again have near identical prints.
If we attach the same 50mm lens to both cameras and take a photo at f/2 of the same subject from different distances so that the subject size is the same and print both images on 6" X 4" paper there will be some noticeable differences. The perspective will have changed because the CB image was taken at a greater distance from the subject. The subject will appear compressed in the CB image compared to the FF image. If background details are visible the background will also appear closer to the subject than in the image from the FF sensor. Because the 50mm lens was focused at a 50% greater distance, the DoF also increased by 50%. If the subject was at 10' using the FF camera and 15' using the CB camera here are the resulting DoF calculations:
- 50mm @ f/2 from 10' on FF: 9.33' to 10.8'. DoF of 1.45' (17.4"). The DoF ranges from 8" in front of to 9.6" behind the 10' point of focus (PoF).
- 50mm @ f/2 from 15' on CB: 14.0' to 16.2'. DoF of 2.18' (26.16"). The DoF ranges from 12" in front of to 14.4" behind the 15' PoF.
These calculations are based on a circle of confusion (CoC) of .03mm for the FF camera and .02mm for the CB camera. This is because we are printing at 600 ppi for the FF and 400 ppi for the CB (and the pixels are the same size for both- 0.01mm or 10µm).
In reality, we all know the pixels on most FF sensors are larger than the pixels on most newer CB sensors. They range from 6.92µm on the 18MP FF Canon 1D X to 7.21µm on the 16MP D4 to 4.7µm on the 36MP FF Nikon D800. The crop bodies go from 4.16µm for the 18MP Canon 7D to 3.89µm for the 24MP Nikon D7100 (the D7200 will be around 3.0µm) to 5.08µm for the 14MP Sony SLT Alpha 33. In all cases the pixel size is considerably smaller than the generally accepted CoC of .03mm (30µm) for FF cameras and .02mm (20µm) for 1.5x CB cameras. For 1.6x CB Canon cameras 0.019 (19µm) is generally used. The largest size pixels Canon has used in the last decade or so was 8.2µm for the 12.8MP FF 5D and the 8.2MP APS-H 1D mkII. What all this means is that at the pixel peeping level, focus blur will be visible even for objects within the accepted DoF because the accepted blur circle is anywhere from 4 to 7 times larger than the pixels on current DSLRs. To calculate DoF at the pixel level you would need to use a CoC the size of your camera's pixels which would be much narrower than most DoF calculators use.