The PS4 controller is supported by Windows, and the game recognizes it just fine. However, the buttons are all mixed up and the Calibrate function does not work at all. Basically, while the calibration function is active, the game does not recognize that any buttons are being pressed.
Now, before anybody points this out, I am already aware that you can edit the FFXIV.cfg file to get the buttons mapped properly. However, I am also reasonably certain that editing game files is against the ToS, so I am reluctant to do so. There are also a couple of 3rd-party programs that cause the PS4 controller to be emulated as an XBOX360 controller (thus bypassing the issue entirely), but to my mind this situation is also not ideal, since the game does recognize the controller, so there is no reason why it should not be able to be calibrated properly from within the game itself.
This isn't an issue with my system. From what I've seen, this happens for everyone who tries to use a PS4 controller with this game (there are a couple of threads on Reddit about this issue and how to fix it). I posted this in Bug Reports but it was closed, and I was suggested to post here, even though I am certain that this is a problem with the coding of the game (or of the Windows driver for the controller, perhaps) and there is nothing any end-users can do about it.