I dont have this issue, but have you tried opening it directly and not through a shortcut? Some times I had the issue of league being stuck on the pre launch screen, and that fixed it.
Maybe try opening one of these:
/Applications/League of Legends.app/Contents/LoL/Play League Of Legends
/Applications/League of Legends.app/Contents/LoL/RADS/system/UserKernel
/Applications/League of Legends.app/Contents/LoL/RADS/projects/lol_game_client/releases/0.0.0.236/deploy/League Of Legends
Im not sure if any of these will work, and some of the links may not be there. I have the Alpha Client so it may have messed with some of my links.
mainly on a mac in order to find these files, open up "Applications" in a finder windows and right click on "League of Legends" than from the drop down click "Show Package Contents" and fish around in that file looking for something to open.
If something works, you may want to drag the icon of what ever worked to your dock so dont have to go fishing for that link every time you want to play.