Close launcher if it is open and ensure that the ffxivupdater (and any other ffxiv process) is not running - if any are still running, end them via the Windows Task Manager (right-click the windows task bar > select "Start task manager".)

Locate the "FINAL FANTASY XIV - A Realm Reborn" identified in the error (as shown in Hanasaku's image) and follow these steps:
Right-click folder > Select "Properties" > Uncheck "Read-only" >Apply to all subfolders (if it asks).

Restart the launcher and (hopefully) it should work