Possible, yes. Sportsmanlike or safe, absolutely not. You can kill a wild hog with a .22LR if you hit him right (just behind the ear). However, a body shot requires something different. How much different depends on the size of the hog. I've killed a hog with a .357 and would feel confident again assuming I have the ability to shoot only the hog I wanted and had the time to place my shot. In my .357 case, the pig weighted only about 80 lbs and the .357 penetated the vitals just fine.
However, an 80lb BBQ sized pig is a far cry from a full grown boar. If you ever skin a wild boar, you'll see the thick shield of gristle that protects his vitals. In the case of a really old trophy boar, this gristle is enough to stop many rifle bullets. You don't want to get in the situation of wounding a mad boar with a pea shooter.
If I'm out hunting hogs, I carry a rifle. If I'm hunting small game in a hog area, I carry a .44mag - even then knowing I would use it with prudence.