Okay, so here's a quick scenario. Humans over the course of time have been selected to breed based on how docile they are. The violent ones are exterminated and all outside sources have been shut out. Could humanity actually forget war and violence?
Theoretically yes, practically no
We get questions like this all the time, "is it possible to remove humanity's violent behavior?" In my answer to how to domesticate hippos I pointed out (among other things) that an experiment conducted on the Russian Red Fox required 20 generations to achieve 35% domestication. Let's assume that some human violent tendencies don't show until adulthood, meaning we need a minimum of 18 years to create another generation or a minimum of 360 years to achieve 35% domestication.
So, after 1,000-2,000 years of devoted effort, you might end up removing humanity's violent temper. Maybe.
Humanity evolved in a competitive environment
Competition is part of our genetic makeup. Yes, theoretically you could create a Bene Gesserit-grade breeding program that would attempt to "domesticate" humanity, but that natural tendency to eat the other kid's school lunch is still there. Foxes (and, I assume, hippos) are nowhere near as sapient as humans — and to get rid of all violent behavior you would have to get rid of greed, envy, jealousy, and every other nasty thought that might lead to, "Jerry's out cleaning his new boat again. He does that on purpose just to rub it in my face! He thinks he's so much better than I am. I think it's time to give Jerry the beat-down of his life!"
But, let's assume that your program succeeds and the competitive nature of our being is removed. No more violence. Well... you'd have to get rid of low self-esteem, too. And shyness. And arrogance... wait...
We actually use some of those behaviors to our advantage
Here's the gist of the problem: if you remove our violent tendencies, you also remove our tendencies to take risks, to innovate, to solve problems, to overcome challenges... all these are just minor forms of violence. How often have I told myself, "I will not be defeated by an inanimate object!" (More often than I'll ever admit to any of you!) Am I justifying violence? Absolutely not. What I'm explaining is that violence is the extreme expression of behavior that is the core of what it is to be human. Arbitrarily remove violence and you remove what it is to be human. The resulting person would be perfectly content to use a stone hammer for the rest of their lives because it would never occur to them to become frustrated over how inefficient it is. ("Rotten hammer!" Wait... that's a violent thought...)
And here's where I need to insert a great observation by John, in comments. If you rob us of our tendency to violence, you affect our fight-vs-flight response — or not. And there's the problem. You either turn us all into cows waiting for the wolves to eat us, or you start killing people because their fight-or-flight response is dialed to fight and you don't want that so you exterminate a normal, useful, valuable response. But, more to the point, no matter how well bred your humans are, you'll never know how they'll react when pushed into a corner until they are. Very deeply rooted, fight-or-flight.
Just for fun, let me put it to you another way. I once golfed with a friend who was studying to become a clinical psychologist. The irony of him slicing into a lake and then throwing his club in frustration wasn't lost on me. But that same emotion that led him to throw his club would lead another to figure out how to stop slicing the ball.... And you want to remove that base tendency.
The real problem isn't violence, it's self-control
This might not sound relevant at first, but a friend of mine once explained that autism isn't a disease or a genetic abnormality per-se. It's not abnormal, so-to-speak. Humans have "settings" for how they see, hear, taste, touch, and perceive the world around them. One of the conditions of autism is that one or more of those dials are set higher than "normal."
In other words — everybody's autistic. Whether or not there's a problem depends on where we arbitrarily scratch a line on the dial that means "beyond this point there be dragons."
Violence is the same. The acceptability of a bold speaker vs. a beach bully is one of subjective acceptance. We admire the bold speaker and vilify the bully — but the base behavior is identical, it's just an issue of how extreme it is...
...or how well we control it.
So, theoretically you could breed violence out of humans, but all those wonderful advances in culture and science would be the last you'd ever see. So it's a bad idea.
Your premise is self-contradictory. If humans are killing violent humans in an effort to wipe out all violent humans, then they would have to eventually kill themselves.
Violence is still violence even if you think of some mechanism for it that somehow nicer than stabbing somebody.
Also, every form of government is ultimately based on a monopoly on violence within a given geographic area (e.g. the police and military provide "legal violence"). It would be impossible to create the penned in area where no outsiders could enter without a very powerful totalitarian regime. Such regimes require a large supply of violent people to stop rebellions and revolutions. These violent people who are part of the state can also attempt to overthrow it, which means you need even more violent people and a system where violent people watch other violent people and apply violence to them if they start applying violence in the wrong places.
The problem exists in exterminating the violent ones. You'd need to use violence to exterminate them.
Failing that, there's an old saying "Those who beat their swords into plow shares become the slaves of those who did not".
The ones who had no problems with violence would react with violence towards any attempt to exterminate them, or worse, alter or eliminate their present and/or future children. By nature, I'm a non-violent person myself, but try to harm my children, and I'd use my full potential.
Therein lies the entirety of the problem. At least some portion of the people would see it as a threat, and respond accordingly.
Committing violence (aggression) and protecting themselves are very different things.
True, violent individuals usually know how to protect themselves, and, conversely, "tough" men are more likely to be violent, but those are conditioned traits.
If we look at the children, we can see how those traits are developing. Some children are naturally more aggressive, and they act this way towards other children, but at first, they have no idea how to defend themselves. If they are becoming a target of other kid's aggression, these little bullies will just flounder. Only later they will learn how to defend themselves, and learn how to pick "soft" targets.
On the other hand, some children are tough like little soldiers, they won't let anyone to bully them - but they are not violent at all. Oh, they do get in trouble for fighting, maybe even more than bullies - but always because another kid started it first.
If we remove all aggressive kids from population, the rest will undoubtedly become softer, because they will have no practice defending themselves. But if the need to defend should arise, those who are naturally "tough" will be quick learners.