I had this exact issue on ps4. I tried resetting router, rebuilding database, powering off ect.

I fixed it by grabbing my smart phone > settings > internet sharing > turn on > share over wifi > on ps4 navigate to network > setup connection > find your phone in the list > enter password the phone supplies(where you turned on net sharing) > Start FFXIV > patch downloaded right away.

*note was on a windows phone.
iPhone and android menus would be different, I cant help you find the relevant menus there.

Whats happening is your phone is acting as a router with its own IP. Not sure why you need a new IP, the patcher for some reason wont download with your old IP from your normal router.
I GUESS you could manually enter a new IP for your router (not obtain automatically) and that would work too...but you have to be somewhat tech savvy, so the above was a much simpler fix for me personally.

I would assume this works as pc fix too, cant be sure though.