The guys around here have been hunting hogs for thousands of years.
The general consensus is bigger is better. From bullets to spears.
Hogs are a nuisance species, but IMO that shouldn't negate general hunting principles. The object is to put it down fast and have them expire quick. Avoid any necessary suffering. And of you do screw up you have an obligation to track that animal and finish it, also to avoid unnecessary suffering.
I've never taken larger game with a .223 I have shot a lot of small game and even some young Roe deer with a .222. I really can't imagine trying to use one on a grown Hog. It is unlikely to be a quick clean kill.
Can you, sure, the question is should you?
If you do screw it up and it is a non fatal shot, the smaller the bullet the smaller bullet hole is likely to pucker shut and the blood trail will end.
It is better to have an exit wound, less likely with a light bullet. A large hole is less likely to pucker shut if you do screw up the shot, the Hog takes off for the thickets and you have to track it.
I've lost one Hog in almost fifty years of hunting them .308-30.06 mostly, I have tracked dozens (maybe a hundred) for other guys who had bad luck or rushed the shot and screwed up. Hogs are tough. The smallest allowable bullet here is 6.5 most often used is 7 mm and larger, the old timers prefer an 8 or 9.3 mm bullet. Bullet weight 150 and heavier, most prefer 180 plus.
Another reason for bringing a bigger gun is you just may get lucky and see a 6-7 year old trophy Boar 300+ pounds. The chances of killing one, other than with a perfect shot, is not likely with pop gun.