On the other hand, an eye for an eye, and soon the whole world is blind. There is no system of checks and balances for the Punisher. He kills who he sees fit to kill.

At first, it would be comforting to know that there is someone out there who is ridding the world of evil-doers. Rapists, killers, child molesters, etc. After a while though, people would come to the inevitable conclusion that his way of doing things has no place in the real world. Unless evidence was left of the crimes they had committed, it would be very difficult to distinguish who "deserved" what they got, so to say. It would also become impossible for him to operate eventually, things like ammo and weapons would be extremely hard to come by once his identity was known.

When all is said and done, I would be against a real-life Punisher, but I wouldn't mind one or two who were similar. The show Dexter is a fairly good example of what would be preferable, in my eyes. The Punisher is too high-profile, and often causes lots of property damage. It would be better to take care of them quickly and quietly, instead of leaving calling cards as obvious as Frank Castle's. 

The world would have to be an even more desperate place before the public would ever approve of someone like Frank. And once they did, you can be sure the federal government would step in and take his place. That road is not one I think any of us would wish to travel down. Furthermore, he would inspire copy-cats, like those seen in The Dark Knight impersonating Batman. This would also be disastrous, as they would not be nearly as meticulous as Frank is.

To summarize: No. Not if they used methods identical to the Punisher's.