According to John Dayton,^1 the last dateable instance of an attempt to settle a war with a duel in Ancient Greece is in 607 BC. After that, the increasing sophistication of government and legal systems, along with the development of formal rules of inter-state conflict arbitration, rendered duels obsolete.
The trouble with the duels we hear about was that the outcome was almost always disputed. Since duelists would often cheat or use trickery to win, their victory was not accepted as a proxy for the victory of their whole tribe or state over another. They would rarely actually settle a war; most of the time, conflict would erupt again soon after, if not immediately after the duel was done. As an attempt to reduce the casualties suffered in wartime, duels were a failure, and they fell out of use as soon as better systems were developed.
A great example of this is the last gasp of restricted conflict in Ancient Greece - the so-called Battle of the Champions, fought around 550 BC between Argos and Sparta (see Herodotos 1.82). Both sides agreed to send just 300 men to fight, and to withdraw the rest of their armies so they would not be tempted to interfere. The result of this "duel" would determine, with minimal loss, who would hold dominion over a disputed patch of farmland between the two states.
The 300 champions on each side fought until only one Spartan and two Argives were left standing. At this point the Argives (Alkenor and Chromios), seeing themselves in the majority, declared themselves the victors and went to Argos in triumph. When they left, however, the surviving Spartan (Othryades) declared himself in possession of the battlefield; he stripped the bodies of the dead and returned to his post, defiant.
When both sides heard what had happened, they both claimed that they had won the battle. Predictably, their disagreement soon escalated, and the full armies of Sparta and Argos fought a pitched battle after all. With hundreds more dead, victory finally fell to Sparta. However, Othryades succumbed to survivor's guilt and committed suicide on the battlefield. It was a pretty bad day for everyone.