Here's how I managed to fix it on my end.

Go to: C:\Users\Rey\Documents\My Games\FINAL FANTASY XIV - A Realm Reborn\downloads\ffxivpatch

Obviously the 2nd line will be YOUR username and not "Rey." Once open, just DELETE the one or 2 folders in there; they will have weird names like 4e9a232b and such. These folders are just where the game is downloading the patch to and will keep everything else (screenshots, ui settings) intact. You can try re-downloading the patch now and unless you have other issues, things will go smoothly.

I had this happen when I first picked up the game awhile ago, and this was how I fixed it then as well. After having to go to great lengths to figure this out, I laughed at how easy it ended up being. Posting this on the SE forums as well, good luck and see you on Gilgamesh!