the obvious suggestion would be to try Raspbian first. If it boots then you'll know it it something to do with RetroPie, rather than the hardware.
if it boots in Raspbian - I advise trying Jessie Lite for the fastest way to cli - and you can run the following, please post the output here:
curl https://get.pimoroni.com/diagnostic | bash
and, if possible:
dmesg > dmesgoutput.txt
... the later should output the boot up messages to a file that you should be able to move to your
/boot partition and post here using a different computer. It may contain some clue about what is going on.