Can't Complete Version Check

[Raist]

I was able to ping all 3 of the servers.

Those are web servers... not even connected to the game. Here's a few DNS names that are hard-coded into the executable, with the IP's they should resolve. See if you can get to them by name:

frontier.ffxiv.com (124.150.157.190)
patch-gamever.ffxiv.com (124.150.157.126)
neolobby02.ffxiv.com (199.91.189.74)

The 124 IP's are servers in Japan, the 199 is a lobby server in Canada.

If basic routing is good, than it may be an issue with your files. Many have reported a mismatch in the version check files. Navigate to this folder:
C:\Program Files (x86)\SquareEnix\FINAL FANTASY XIV - A Realm Reborn\game

Open the ffxivgame.bck and ffxivgame.ver in a text editor to see if they have version numbers in them. If they match, then your patch files could have become corrupted. You can find/remove the downloads in here (insert your systems user account name at <USERNAME>):
C:\Users\<USERNAME>\Documents\My Games\FINAL FANTASY XIV - A Realm Reborn\downloads\ffxivpatch