As Utanapishtim tells Gilgamesh "No one can see death, no one can see the face of death, no one can hear the voice of death, yet there is savage death that snaps off mankind" (10. 290-293). Is it any surprise that Gilgamesh doesn't end up being successful in his battle against such a mysterious, invisible antagonist? Though of course, you've got to hand it to him for trying. And yet, isn't Death itself just a front for another, more hidden group of antagonists?
As Utanapishtim tells Gilgamesh about the creation of the world: "After Enlil had pronounced the blessing, the Annunaki, the Great Gods, assembled. Mammetum, she who fashions destiny, determined destiny with them. They established Death and Life, but they did not make known 'the days of death'" (10.305-309). In other words: if Death is Gilgamesh's antagonist, and the gods created Death, doesn't that mean that the gods are Gilgamesh's true antagonists?
If this doesn't sit well with you at first, just think about all the times Gilgamesh comes into conflict with the gods over the course of the story. When he and Enkidu are doing battle with Humbaba, Enkidu urges him to kill the monster quickly, before the gods find out. Later, Gilgamesh rudely insults a goddess (Ishtar) who offers herself to him, and he and Enkidu kill the Bull of Heaven. We won't blame Gilgamesh for throwing the Bull's leg in Ishtar's face (that was Enkidu's doing), but doesn't the company we keep say something about us?
Of course, Gilgamesh isn't the enemy of all the gods; he's obviously not against his mother, Ninsun, and the god Shamash has his back. Still, we definitely think there's some conflict going on between Gilgamesh and the divinities. Or maybe, it isn't even the divinities so much as what the divinities represent: fate.