I'm having this same issue. it works fine on PS3 and the PS4 beta is running fine, but i can't get it to work on my PC. It was working fine on Friday, 28 February, then when i went to launch on 1 March I started having issues. For instance when I launched it the first time it said it could not verify patch notes.

In order to fix this, I dropped the Windows Firewall from the control panel. Which fixed that issue.

But then it got stuck and said it couldn't complete the version check. Just like your problem
So I did what I did before and dropped the Windows Firewall and this time I turned off Antivirus for an hour as well.
I also ran FFXIVboot.exe as an administrator (right clicked the executable and selected "Run as Administrator"). I also, under the PC's admin account, unset the Square-Enix Folder's Read-Only setting (right-clicked and went to properties). Even as an admin the properties kept resetting to be read-only.
I've tried uninstalling and reinstalling and I've tried resetting the computer a few times. Nothing is working.