To test, navigate to your program files (x86)/Square Enix/FFXIV folder, right click and run ffxivboot.exe as administrator. Make sure all of the files in that folder are running as administrator.

If you haven't already, completely uninstall IE, run CCleaner and then re-install it. If that doesn't work, also try re-installing FFXIV. In addition, make sure that AVG didn't quarantine any of your installation files by mistake. It's also possible that windows UAC blocked something. Turn your UAC to off or low, and make sure you have full administrative rights for everything if you re-install FFXIV.
I went ahead and uninstalled the game first and reinstalled all the files as administrator, double checking to make sure they are.
No fix.

I uninstalled IE, ran CCleaner and reinstalled.
No fix.

I checked my quarantine files and I had 0, and disabled AVG during the fresh install of FFXIV. No fix.

I have had windows UAC turned off since I built the rig 8 or so months ago.
No fix.

Rentahamster

You sure you're not running some sort of P2P IP blocking software or something? Look on your list of processes and turn off everything that is non-essential.

Just resetting your IE settings to default doesn't fix this?
I do not have any P2P IP blockers running or even installed since I don't torrent files anymore. No fix.

I ended all non-essential processes before the fresh install and after the reboot.
No fix.

I have reset my IE settings to many times to count.
No fix.

This has to be the most frustrating problem I have ever encountered with a computer, even worse then printer problems...lol