The cartridge can matter, but perhaps less than some people think.
There are four ways in which shooting an assailant actually can stop a fight:
- psychological -- "I'm shot, it hurts, I don't want to get shot any more."
- massive blood loss depriving the muscles and brain of oxygen and thus significantly impairing their ability to function
- breaking major skeletal support structures
- damaging the central nervous system.
Depending on someone just giving up because he's been shot is iffy. Probably most fights are stopped that way, but some aren't; and there are no guarantees.
Breaking major skeletal structures can quickly impair mobility, but someone with a gun can still shoot. And it will probably take something bigger than a .22 or .32 to reliably break a large bone.
Hits to the central nervous system are sure and quick, but the CNS presents a small and uncertain target. And sometimes significant penetration will be needed to reach it.
The most common and sure physiological way in which shooting someone stops him is blood loss -- depriving the brain and muscles of oxygen and nutrients, thus impairing the ability of the brain and muscles to function. Blood loss is facilitated by (1) large holes causing tissue damage; (2) getting the holes in the right places to damage major blood vessels or blood bearing organs; and (3) adequate penetration to get those holes into the blood vessels and organs which are fairly deep in the body. The problem is that blood loss takes time. People have continued to fight effectively when gravely, even mortally, wounded. So things that can speed up blood loss, more holes, bigger holes, better placed holes, etc., help.
So as a rule of thumb --
- More holes are better than fewer holes.
- Larger holes are better than smaller holes.
- Holes in the right places are better than holes in the wrong places.
- Holes that are deep enough are better than holes that aren't.
- There are no magic bullets.
The bottom line is that a lower power cartridge with a smaller caliber bullet will make smaller holes and may not be able to as reliably penetrate to where those holes need to be to be most effective. On the other hand, a small gun that you actually have with you and that you have trained with and can manage well will serve better than the larger, more powerful gun that is so large you left it home or so powerful you can't shoot it accurately.
So it comes down to a compromise. If you can learn to manage a more powerful gun and find ways to keep it with you and handle it effectively, you're likely to be better off.