I had this happen to me when I stopped it halfway through downloading a patch, but the fix turned out to be quite easy so hopefully it's the same for you:

Navigate to this path: My Documents\My Games\FINAL FANTASY XIV - A Realm Reborn\downloads\ffxivpatch

Inside that folder will be probably be 1 or more subfolders with 8 character alphanumeric names.

Simply delete all the folders within that directory and then re-run FFXIV and let it retry patching and it'll hopefully work for you this time!