Grey market in this case is USA stock sold by unauthorized dealer to go around the Canon minimum advertised price (MAP). So the USA warranty card is still included. It is not the same as Japanese stock sold in the US. Obviously Canon can use the serial numbers to see who the original dealer the camera was delivered to. Whether the repair center does anything with it or not is the question.
I had my 5D Mark II and 24-105L lens sent in for out of warranty repair. They will fix any Canon gear as long as you pay for the repair. So if you bought this, Canon will still repair it for you if needed. It is just a matter of how much you would need to pay to get it repaired. For me, I dropped my 5D Mark II on marble floor with the lens on it. The prism housing cracked. The repair was $450. The lens was fine but I sent it in for clean up and check up. That cost $250.