If I had to hunting a deer with a sling (keeping in mind I know nothing about hunting) this is what I would try.

I´d get a chicken sized egg or a slightly bigger projectile and I would sling as hard as I could aiming for the neck.
I´d go for the neck because I think that would be the most vulnerable spot for blunt force trauma. The neck is made from individual vertebrae, none of which are easy to break, but the soft joints between the bones are not too tough to harm. So a big rock traveling at terminal speed might dislodge at least one vertebrae. Maybe that could make some damage to a large mammal, at least leave it paralyzed.

http://www.youtube.com/watch?v=wwNbwPVFkSc&feature=related

Some awesome video I found just now. Kangaroo vs. Atlatl........kangaroo looses.

If you threw a rock to one of those, unless you got a perfect shot at the neck, maybe it won´t do much good. But with the atlatl, the hunter gets a nice, clean kill.

Back to the kinetic energy discussion, any kind of projectile has an optimum velocity and weight for maximum kinetic energy.  For weapons whose energy is supplied by human muscle, that optimum point is dependent on the individual.  For more mechanical weapons, say cross bows on up to firearms, more general statements can be made.  I included crossbows, since many have mechanical cocking aids, so pretty much anyone who knows how can use the heaviest cross bows.

If I missed this in the discussion my bad, but how close you are to the deer would make a huge difference.   If I can get within 35 yards of a deer and get off a shot without spooking it and hit the deer in the neck with a large egg sized stone, it's going down.  I'm not saying I'd kill it outright, but it will be stunned hopefully long enough for me to dispatch it with a club or survival knife.  Of course there are a few givens here that might not necessarily happen.  Getting within 35 yards of a deer is easier said than done.  Deer are super alert and not spooking one even with Apache style would be most difficult, and finally managing a neck shot with a sling from 35 yards is difficult.  What I'm saying here is that I firmly believe that there's enough power in the first 35 or so yards of my sling projectile's path to bring down a deer, but the power involved isn't the only factor.  You'd have to be very good and very lucky to get such a shot off effectively.
    After 35 yards or so there's a big drop off in power and over fifty yards the stone looses a lot of its pizazz and sort of trails off, even if it's going to travel much farther.  Your best bet to get a deer with a primitive weapon is by far the bow and arrow, IMO.  It's easier to use, you don't have to wind up to shoot it, it's more accurate, and the projectile is just as dangerous throughout it's entire path.  I can't comment on the atlatl because as of yet I've never used one.  Hopefully that'll change someday soon.