D or later. The G-type lenses transmit the distance information too. As it is meant only for internal calculations for flash exposure, Nikon don't see a use for displaying the value.
The DOF presented in the iPhone apps is a very approximate value, however many decimals the app presents, as the CoC used in the calculations is selected using a very rough model. To get predictable results, you have to shoot more and learn how each lens behaves in different situations. You will get wildly different pictorial results if you use a 14-24, 24-70, 24/1.8 or 24/1.4 even if you set them all to 24 mm, focus them on the same point and use f/2.8. Besides the reproduction scale and the viewing distance, the character of the transition from sharp to unsharp has a huge impact on the perceived DOF.
The distance is, by the way a secondary factor... The primary is the magnification scale, not the distance itself. The scale is calculated using the distance and focal length. With an internally focusing or close range corrected lens, the focal length is shorter than the nominal as soon as the lens isn't focused on infinity.