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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.