I think there are too many people in the world, and too much anonymity. A duel, and the honor it supposedly satisfies, are based on your reputation, your standing within the community and the world you live in. But unless a celebrity dueled, nobody would know who the combatants were and nobody would care, rendering the whole idea of a duel pointless.
In order for duels or other contests of honor to work, you need a small community where everyone knows and cares about each other, so the duel, accepting or declining it, and winning and losing it would actually mean something to the people you live with and rely upon. This happens in smaller, isolated communities (like towns, villages, and tight knit cities), or in places with exclusive social classes (in most cultures duels are something that only nobles do, because there are relatively few of them and they're easier to keep track of than the thousands of faceless peasants).
That being said, I think trial by ordeal is a concept that we could use a dose of in this modern age. Duels are part of a larger category of trials by ordeal, and basically with all such trials the idea is that, in order to make a legal claim, you must risk pain, injury, and/or death to prove that you are sincere in your claim and that it's important to you. I saw a documentary about this village in Africa where, in cases when it was one person's word against another's and there were no other witnesses or evidence to draw upon, both the people involved in the claim would have to reach into a bowl full of boiling hot oil and retrieve a metal ring. Since both people had to do it, it prevented frivolous claims--imagine if every time you went to court to fight a well-deserved traffic ticket you had to stick your hand in boiling oil! It also tested the resolve of the people--if you're innocent, your indignation and determination to defend yourself will likely be much higher than someone just trying to get a few bucks out of it. Or at least that's the idea.
The problem with these, and with duels in particular, is that people can gain an unfair advantage by becoming good at facing the ordeal. In order to be effective a duel should take place using weapons that both parties are equally skilled (or unskilled) with. In many dueling traditions the choice of weapon usually went to the one who was challenged, meaning that a skilled swordsman couldn't just go around challenging everyone to a swordfight--anyone he challenged would pick something other than swords. During the classic dueling time periods it was assumed that everyone was comfortably familiar with swords or pistols, so these worked as a good equal playing field in most cases. But there was also an element of bravado to it--someone looking to earn additional cred might choose the weapon their challenger was strongest with in order to "defeat them at their best!" But the basic idea is that you're supposed to risk your life in order to demonstrate your honor and resolve that you are in the right. It's basically a big game of chicken.
Also, even in very duel-happy cultures, a duel was always the last resort after more traditional legal options had been exhausted. You don't just jump immediately to a duel to the death over trivial issues. It was usually either an unresolvable case of one person's word against another's, without any other witnesses or evidence, or it was a matter of honor and reputation, where the law really can't impose fines or awards (the court can't order your community to respect you, after all). And even with duels there are different grades of seriousness--most people think duel to the death when they hear the word duel, but there were duels to first blood (the person who got cut first lost), and in a lot of pistol duels the requirement was that each party had to take a shot at the other--even if they both missed, they both risked their lives to prove their honor, and therefore their reputations were established and nobody would doubt their bravery.