What version of OS X do you have? I am having the exact same issue, no error, no dialog, nothing happens after I double click the app.
I then poked through the forums and system requirements, and noticed that we need OS X 10.7.x or higher, which means Lion+, unfortunately, I have snow leopard(10.6.8), so I’m guessing that is the issue. It’s a shame that GW2 requires 10.7+.