Of all the varied and creative heroic spirits in the Fate series, two of the mega franchise's most popular are both from the original 2004 visual novel. The first of these is the elusive and mysterious Wrought Iron Hero, known simply as Archer, while the second is the King of Heroes himself, Gilgamesh. Both of these heroic spirits have similar abilities, causing them to be compared to each other quite often.
But, as the title gives away, a duel between Archer and Gilgamesh is simply not counted among Fate/stay night's many epic battles. Deprived of an official showdown between these two characters, fans can only speculate as to who would win between them. For clarity's sake, we'll only be looking at Archer and Gilgamesh as they appear in the original Fate/stay night visual novel, as well as its anime adaptations. Naturally, heavy spoilers, particularly Archer's true identity, lie ahead.
10 Gilgamesh: No Master
Because of the events of Fate/Zero, Gilgamesh exists in the world of Fate/stay night as an independent servant. That means that, unlike other servants, he has no master. While this is an inconvenience in many ways, such as no longer being able to turn into spirit form, it is an advantage in battle.
As demonstrated in Fate/stay night with Saber's weakened abilities, the capabilities of a servant's master contribute a great deal to their battle capacity. Unlike Archer, this isn't something Gilgamesh has to worry about. Yes, Archer does have independent action, which allows him to remain masterless for a time, but his mana depletes quickly in this state, with no way to replenish it.
9 Archer: The Law Of Transitive Property
The true name of the Wrought Iron Hero is heroic spirit EMIYA. In other words, he is a future version of protagonist Shirou Emiya. This is important to note because it contextualizes Archer's abilities, as he is basically a version of Shirou with maximized potential.
Similarly, this is important because, while Fate/stay night lacks a proper duel between Archer and Gilgamesh, it does have one between Shirou and Gilgamesh. And in this duel, Shirou is victorious. It should stand to reason, then, that if Shirou can beat Gilgamesh, then by applying the law of transitive property, it can be assumed that Archer also beats Gilgamesh, since he is effectively Shirou but better.
8 Gilgamesh: The Law Of Transitive Property
However, the law of transitive property is fickle. For example, when Shirou is fighting Gilgamesh, he comments that he'd probably lose to nearly any other heroic spirit, even with Unlimited Blade Works. A duel is a contest between skill, not raw power. Therefore, the law of transitive property cannot apply.
Archer is the same person as Shirou, but they're not identical. Additionally, just because Shirou beat Gilgamesh once doesn't mean that he'll beat him every time. If the focus is put squarely on Archer and Gilgamesh's respective abilities, then Gilgamesh's chances of winning improve drastically.
7 Archer: Gilgamesh's Fear
It should also be noted that Gilgamesh, though he does his best to hide it, is actually scared of Archer. Obviously, he never comes out and says it, but his actions towards him in the Unlimited Blade Works route make this clear. He is uncharacteristically agitated by his very presence, and routinely calls him a "faker."
This is because Gilgamesh knows that Archer's Unlimited Blade Works is the perfect counter to Gate of Babylon. When Archer threatens Gilgamesh, he takes the threat at face value and doesn't retort, which is also jarringly uncharacteristic of him, but it confirms just how much begrudging respect he has for Archer.
6 Gilgamesh: Gilgamesh's Respect
While Gilgamesh is scared of Archer, on the flip side, this means that he holds in relative high regard. He acknowledges Archer's combat ability, at least internally, and thus sees him as a threat. Shinji may not take Archer's threats to heart, but Gilgamesh sure does.
Potentially, this could backfire on Archer, as Gilgamesh holding such a respect for him could cause him to go all-out. The last opponent Gilgamesh went all-out on, Fate/Zero's Rider, paid the ultimate price for earning the King of Heroes' respect. Potentially, Archer could repeat this error.
5 Archer: Enkidu, Chains Of Heaven
One of the greatest noble phantasms housed by Gilgamesh's Gate of Babylon is Enkidu, the Chains of Heaven, named for Gilgamesh's one true friend. It is a binding chain that grows more dangerous in proportion to the divinity of their target.
But herein lies the problem. It is demonstrated in Fate/stay night that Archer is not only not divine, but is actually so distorted that the mud of the Grail has no effect on him. To Archer, the weapon that is Gilgamesh's trusted ace-in-the-hole is no more than a regular, unintimidating chain.
4 Gilgamesh: Armor
An important stipulation to make regarding Shirou and Gilgamesh's fight is that Gilgamesh wasn't wearing his armor. This is because Gilgamesh didn't respect Shirou as a worthy opponent, instead seeing him merely as a bug to be squashed.
Conversely, Gilgamesh would be much less likely to make this sort of mistake when battling Archer. While it's definitely possible that his pride could get the better of him, it nonetheless remains an obstacle that Archer would have to take into account. If Gilgamesh were wearing his armor, it'd be much harder to deal significant damage to him.
3 Archer: No Armor
However, regarding Gilgamesh's decision as to whether or not he should wear his armor, it must be considered just how infrequently he actually did so in Fate/stay night. Saber was the only opponent he routinely considered worthy of dueling in his armor. If not even Berserker, the legendary Heracles, was worthy, is it really reasonable to assume that Archer would be?
Of course, if this were Gilgamesh at the time of Fate/Zero, things might be different, as he wore his armor for every one of his fights. But this is not the version of Gilgamesh that's being dealt with, so this argument doesn't stand.
2 Gilgamesh: Ea
The final and most significant obstacle for Archer to overcome would be Gilgamesh's greatest sword, Ea. It is the only sword that Unlimited Blade Works cannot reproduce, which is really saying something, considering it can reproduce Excalibur.
But its greatest strength lies in its ability to divide space. As Gilgamesh so wondrously demonstrated in his final duel against Fate/Zero's Rider, Ea has the ability to totally deconstruct reality marbles. This would completely negate Archer's greatest advantage against Gilgamesh, drastically lowering his shot at winning. It should be noted that, in Shirou's fight, he was desperately struggling to fend off Gilgamesh before invoking Unlimited Blade Works.
1 Final Verdict: Archer: Unlimited Blade Works
All of Gilgamesh's advantages, particularly Ea, hinge on him being able to utilize them effectively, as he normally would. But battling a heroic spirit in their own reality marble does not constitute normal circumstances. Gilgamesh was lucky when it came to Rider's Ionian Hetairoi because he had all the time in the world to prep Ea.
Such leisure would not exist in Archer's Unlimited Blade Works. The advantage provided to Archer by his noble phantasm is the ability to completely overwhelm Gilgamesh and his Gate of Babylon, effectively rendering it useless. Thus, Gilgamesh probably wouldn't even have the time to use Ea. Archer would cut him down before he even got the chance.