An epic hero is a character posessing exceptional/extraordinary/superhuman qualities, such as extreme strenght, ability, intelligence, possibly supernatural powers, as well as an exceptional moral status (unlike a tragic hero, he is not fundamentally flawed). He may be of noble or even godly descent, he participates in a journey or a quest and returns home significantly transformed. He is an embodiment of traits most valued in the society, through the deeds he performs and the morals he stands for.
Now, let's look at Gilgamesh.
He is two-thirds god and one-third man: so check for godly descent.
He is king of Uruk: check for noble descent/status.
He is opressing his people, overworking the young men and claiming the right to sleep (not consensually, we can assume) with all brides in the land on their wedding night. People are crying to the gods in desperation because he is such a cruel ruler- that's where he is non-heroic; he doesn't fit the profile of the noble, brave, just ruler you would expect an epic hero to be. (Hence the gods creating Enkidu and their fight breaking up specifically as Enkidu prevents Gilgamesh from entering a maiden's wedding chamber)
He defeats Enkidu and they become friends: another heroic characteristic; he has demonstrated a feat of strenght.
Gilgamesh decides to go slay the demi-god monster Humbaba (with blessing of his godess mother, no less): clearly heroic; the beginning of a quest
As he approaches the Cedar Forest, Gilgamesh experiences fear/has dreadful dreams and Enkidu conforts him: foreseeing the future through dreams is a heroic characteristic, but at the same time he shows very human/ not-so-heroic vulnerability; could it be that his friendship with Enkidu is what makes him more human than hero, and hence morally relevant/relatable?
Gilgamesh shows mercy to a defeated Humbaba, but eventually kills him, as an enraged Enkidu demands; this one is quite debatable: did Gilgamesh take the morally right decision? Did he show heroic mercy, but as a flawed human being, with a weakness for his friend, succumbed to violence? Or did he show human mercy, but heroic loyalty and courage when killing the monster?
Gilgamesh refuses the advances of godess Ishtar: this shows heroic courage, justice and endurance/chastity; however, the premise of 'enraging the gods' hints more towards a tragic, rather than a heroic story.
Gilgamesh and Enkidu slay the bull of Heaven: typical heroic deed
Enkidu dies, as per desire of the gods; Gilgamesh laments his death: again, that's really interesting to discuss: is Gilgamesh responsible for Enkidu's fate?
Gilgamesh starts the quest for eternal life: typically heroic
Gilgamesh kills the stone-giants out of spontaneous rage: surely NOT heroic; a moment of human weakness (but one that is often part of the character developement of epic heroes: compare Achilles in the Ilyad).
Gilgamesh fails to stay awake for six days and seven nights: that's where he learns himself the limits of his heroic status, he is not invincible. Same theme followed through with the plant of youth being stolen by the snake.
You could argue that (1) Gilgamesh's heroic status is limited by his heroic condition and (2) he doesn't start off with the heroic superior moral status, but gains it through his friendship with Enkidu