Hmm, that is surprising. For Spine to show the launcher window, it should have written the spine.log file. What happens if you open Finder, press shift+cmd+G (I think that is right, not in front of OS X right now), then enter ~/Library/Application Support/Spine? If that doesn't work you can try running Console. When you run Spine the logging should show up in Console, though possibly mixed with other logging.
Please post the whole Spine log once you find it (not just the error). The hope is it sheds some light on exactly why Spine is unable to reach the server. However, if the log just says it can't connect, we'll need to figure out why. OS X has a firewall, though users haven't had problems with it blocking Spine in the past. There are multiple Spine servers in different geographical areas, so it is unlikely to be a problem with your ISP being unable to route traffic to a server.
Another thing to check is that you are running the latest launcher. To update the launcher you'll need to download it again from your Spine license page and reinstall.
