He can. It'd be freaking lousy roleplaying, but he could.
Realism arguments tend to fail to persuade me, where D&D is concerned. D&D is not trying to model reality. It's trying to model heroic fantasy. Things like falls really shouldn't be fatal to a powerful fantasy warrior...try to remember the last heroic fantasy novel you've read where the bulging barbarian hero slipped, fell down into a chasm and died. The end. Oops.
Well, obviously a story hero won't, because the narrative won't let him. If he did, story's over. But RPG's introduce randomness into the mix, so there's suspense and surprises. This also means though, that the heroes of the story (PC's) can die for non-plot reasons. Good for suspense, bad for story. So they compromise.
They made heroes larger than life, and biased the world so it's not as dangerous as it realistically should be. The result is that the heroes are HARD to kill...much harder than is realistic. That means players aren't afraid to take risks (like making that Climb check to go over the rope over the chasm) because their characters are tough enough to survive a bad dice roll or two. At the same time, it means the heroes aren't constantly requiring raising or replacing...which screws up stories and plots.
If you want brutal realism, try a low-point game of GURPS.

Still very fun, but it favors a -hugely- different playstyle.