I know some ancient history enthusiasts have discussed at length whether Roman army tactics could defeat a Greek phalanx (we're split on that one - some think the maneuverability of the Roman maniple could outflank a phalanx formation but others think it would depend on the topography of the battlefield.) Of course, from the descriptions of the Greeks battlefield behavior in the Iliad, the Greeks at the time of the Trojan War didn't fight in the disciplined phalanx that was used later in Greek history.
But, history itself shows that the Romans repeatedly defeated phalanx formations:
The first encounter between a Greek phalanx and a Roman legion was the battle of Heraclea in 280, in which Pyrrhus of Epirus overcame his Italian enemies, but suffered heavy losses because the Roman army was more flexible and could replace the soldiers in the first line; they could continue to fight much longer. This flexibility was Rome's main advantage, especially when rearrangements had to be made during the battle - something that was always necessary during a fight on a hilly terrain. In June 197, at Cynoscephalae, the Roman commander Titus Quinctus Flamininus overcame the Macedonian king Philip V, and the Greek historian Polybius of Megalopolis concluded that this battle was the best example to show that legions were superior to the phalanx (World History, 18.28-31). - Courtesy of Livius.orgImage of a Secutor (standing) fighting a downed retiarius courtesy of Wikipedia.As for a mano-et-mano confrontation between a seasoned Roman-trained officer and a Greek hoplite skilled in the use of a thrusting spear, I'm not so sure myself, although it appears that the mighty Maximus is about to be victorious in this clip. Achilles would have a much greater reach with his spear but a full panoply of his own time period - a breastplate, greaves, hoplon, and a tunic of stiffened linen - would weigh as much as 15 kg, Maximus is dressed in only a ragged tunic and leather armor and armed with only a short gladius. Maximus would have speed and agility on his side like a retiarius but without the net and a longer weapon.
Russell Crowe and his character Maximus are admired as physically fine examples of an alpha male in our society but the Romans would have interpreted his lack of armament as a symbol of a demeaned status, even more than a slave.
The more skin left unarmoured and exposed, the lower a gladiator's status and the greater his perceived effeminacy. Likewise, the engulfing net may have been seen as a feminine symbol. The light arms and armour of the retiarius thus established him as the lowliest, most disgraced, and most effeminate of the gladiator types...
...There is evidence that those net-men wearing tunics, known as retiarii tunicati, formed a special sub-class, one even more demeaned than their loincloth-wearing colleagues.The Roman satirist Juvenal wrote that:
So even the lanista's establishment is better ordered than yours, for he separates the vile from the decent, and sequesters even from their fellow-retiarii the wearers of the ill-famed tunic; in the training-school, and even in gaol, such creatures herd apart….The passage suggests that tunic-wearing retiarii were trained for a different role, "in servitude, under strict discipline and even possibly under some restraints". Certain effeminate men mentioned by Seneca the Younger in his Quaestiones naturales were trained as gladiators and may correspond to Juvenal's tunic-wearing retiarii. Suetonius reports this anecdote: "Once a band of five retiarii in tunics, matched against the same number of secutores, yielded without a struggle; but when their death was ordered, one of them caught up his trident and slew all the victors." The reaction of Emperor Caligula showed the disgust with which he viewed the gladiators' actions: "Caligula bewailed this in a public proclamation as a most cruel murder, and expressed his horror of those who had had the heart to witness it." The fate of the retiarii is not revealed. This was probably not a standard competition, as real gladiators did not surrender so easily. Rather, such tunic-wearing net-men may have served as comic relief in the gladiatorial programming. - Wikipedia