It is, of course, possible to prove pretty much anything using statistics, but I doubt this is one of those cases. Here's the basic A/B comparison:
Smacking/spanking/etc allowed: crime rates/etc value at year XXXX is "X"
Smacking/spanking/etc banned: crime rates/etc value at year YYYY is "Y"
The years and their respective values are not numbers in any way controlled by the lawmakers. They are yearly statistics. They are also numbers that can be obtained in similar manner from other countries, as reported by the other country.
Now, if Y<X, I'm sure you agree that in principle it means the new legislation works. Note that since they avoid the issue of defining "smacking" as opposed to "hitting" by making them equal for the purposes of the new law, any smacking/spanking/etc is a crime and thus means every reported instance after the legislation was passed affects Y.
Of course, there is any number of other reasons for crime rates fluctuating so getting relevant numbers is hard. What causes what? Does one change correlate to the new figure or not?
So the easiest solution is to interpret the numbers using the same methods for both X and Y. With everything else being equal, the easiest explanation (attributing the smallest change) is likely (or at least possible) to be true.
It's not perfect but it's what we have.
What's the alternative? Well, you could say that "no, I don't believe 'smacking' is harmful to my child" and be right, but in a situation where the laws also are permissive enough to allow the person who *hits* his child or smacks it with intent to harm to go unpunished, the statistics will be skewed, but for you it will be in the wrong direction for what you were trying to prove.
So your burden of proof is actually much harder than mine. For my study, I have been able to say "ban it all" and watch what happens, but you have to a) define "smacking" as opposed to "hitting", b) provide the statistics that support you definitions, and c) interpret them in such a way that a comparison between them and my study is possible and relevant.
See the difference? If I choose to ban every form of smacking, regardless of definition, my work is easier, with everything else being equal, and I actually don't have to work the numbers as hard.
If we assume that both parties might work the numbers in their favour but the basic data is freely available, it should be easy to make a feasibility study.
What does this mean? Well, only that I do have the numbers and an accompanying study, but you don't. If you mean to prove me wrong, you'll be busy for quite some time.