Stopping power of any given round is a function of its speed, rifling, weight, caliber, deforming characteristics, and to lesser extents its shape and composition.
A bullet’s stopping power is basically its ability to effectively transfer its energy to the target causing damage. Let’s take each of the characteristics in turn:
Speed – The faster the bullet the more energy it has. Therefore the more energy it can transfer. The downside is that the quicker the bulet, the less time it can spend “in” the target. The target may not be able to contain the bullet, in effect allowing the bullet to pass through the target without transferring much of its energy. .22s move quicker, generally that .45s.
Rifling – the amount of rotation a round has when it leaves the barrel of the gun. The quicker the rifling, the more the round “drills” into the target. The drilling action allows the bullet again to pass through the target without transferring much energy.
Weight – The heavier the bullet, the more energy it carries (gives the bullets being compared are moving at the same speed). This is easy to prove: a heavier round needs more powder to propel it as swiftly as a lighter round. Hence, a heavier round has intrinsically more energy to transfer to the target.
Caliber – The size (diameter) of the round. The more surface area of the bullet, the better able it is to transfer its energy to the target.
Deforming characteristics – The most important factor in transferring energy. Bullets are designed to deform after hitting a target. In effect going “splat”. Hollow-point bulets are perfect examples. They are designed to pancake out after penetrating the target, braking the bullet within the target and transferring just about all its energy to the target (some heat energy would be lost though).
Shape – If the round doesn’t deform, its shape would determine how much energy a bullet would transfer. A needle-nosed round would transfer less energy than a flat-nosed round.
Composition – Lead goes “splat” easier than copper or bronze. Copper jacketed rounds “spit out” their lead innards. Telfon-coated rounds (designed to penetrate body armor) will just drill a nice, neat hole through its target if it is unarmored.
So you see Karl, the ammunition manufacturer has a lot of variable to deal with when developing bullets.
In general, a .45 round has better stopping power than a .22 “if they are both moving at the same speed” because it’s heavier, has more surface area and a larger round will deform easier. In order to answer your question we are not bringing into play composition, rifling, deforming characteristics or shape, or assuming they are the same for both the .22 and .45 rounds.
IRL, of course, this would not be true. The .22 would be moving a lot faster, wouldn’t be jacketed, wouldn’t mushroom or pancake and would have a tighter rifling each of which would significantly affect it’s stopping power.
Here’s a quick rule of thumb: with a .22 you need a head or heart shot to stop someone from charging at you. You hit 'em in the stomach with a .22, they’ll keep coming.
If you get winged in the shoulder by a .45, you probably be spun around and knocked down.