From a web browser on the same computer try going to:

https://frontier.ffxiv.com/

You should see the FFXIV logo and a "page not found" message if it's loading correctly, but that's the hostname that the launcher is trying to access. If you can't see that it in a normal web browser then at least you know something else is blocking your computer from accessing it.

If you can see that page though, try going into Internet Explorer's options (even if you use another browser normally) and deleting the Temporary Internet Files (the cache) and see if that helps (the launcher uses IE).