I recently picked up Shadow of Mordor becaus eit looked like a fun time. But after downloading it and launching the application I was met with a black screen for several seconds and then the game closed and I was put back on the GOG window for launching it. I repeated the process numerous times and uninstalled and reinstalled the game only to be met with the same results time and time again. I was wondering if anyone knew of possible solutions?

You should provide more information about your system (hardware, OS, etc.) in order to get help.

That said, try launching the game using its own executable (ShadowOfMordor.exe), without Galaxy running. Galaxy isn't required and might interfere somehow: at least you might get some useful crash report.
Moreover I suggest you to read this article in the official support website: it's quite generic, but it might contain some useful information about the issue and how to solve it.