Close the launcher, then load up Internet Explorer (even if you use a different browser normally) and then go into the options, and then delete browsing history. Specifically you need to delete "Temporary Internet files and website files" and make sure that "Preserve Favorites website data" is NOT ticked (because if you have a bookmark to any finalfantasyxiv.com page it won't delete the cached data you're trying to delete).

The launcher is essentially a mini Internet Explorer website and if the IE browser cache is corrupted you can get this, but the few times I've had it, clearing the cache always fixes it