One thing to try is to install Wireshark or a similar tool and sniffing the network activity of the launcher.
It should make 2 HTTP connections - one to a server to check the version (to a URL including "win32" and the full launcher version string). That request will normally get error 204 ("no data") but include an HTTP header with the latest version.
The second is the link I posted earlier; it should get a response for that.

If you don't see any response coming in, ISP/network/firewall are prime suspects. If it gets an HTTP error, that error may give insight as to the cause (especially if a direct IE connection doesn't have an issue). If it gets no error, but the launcher still shows nothing, it's probably the launcher itself messing up (in which case, try renaming "C:\Users\<username>\Documents\My Games\FINAL FANTASY XIV - A Realm Reborn" so that the launcher doesn't start out using pre-existing settings).