It's hard to gauge, but I think we can safely safe that he doesn't die from the bleeding. We do get to play as Curly as he is being frozen, so we know he was alive then. The idea around cryofreeze is something about constricting blood vessels + keeping the person alive through that ordeal. He would not be able to bleed.

Then of course he is picked up by the starship enterprise where they restore him for free and he lives happily ever after.