I am having the exact same issue on my mac.

I've downloaded the software 3x now and each time, after I download and put the application into my applications folder and try to launch, the software verifies and then nothing happens. The weird thing is that the mac thinks the software is running, because it will not let me drag the app to the trash. The software however can not be force-quit as it does not show up in the force-quit application box. The only way to remove the software is to restart the computer and then trash.

So, right now, I have no way at all to play FFXIV which is incredibly frustrating. My mac spec's are below as well.

OSX v 10.9.5 (not Yomemite)
Macbook Pro, Retina, 15" Late 2013
2GHz Intel Core i7
8GB 1600 MHz DDR3
Intel Iris Pro 1536MB