How did the Epic of Gilgamesh influence western civilization?

One striking position, particularly expressed by scholar M. L. West, is that a specific version of the epic (which was told in various forms) was a direct source for key characterizations and plot elements of Homer's Iliad and Odyssey. In his book The East Face of Helicon, West concludes:

Both Iliad and Odyssey show, beyond all reasonable question, the influence of the Gilgamesh epic, and more especially of the Standard Babylonian version of that poem . . . This is the version known from Asshurbanipal's library at Ninevah (p. 587).

In particular, he argues that:

...in the case of Achilles, we are faced with the inescapable fact of a special relationship with one particular oriental text: the Gilgamesh epic. The parallels are too numerous and specific to allow any other explanation. (p. 401)

As for the other Homeric hero, "Odysseus' wanderings" are the "strand in the poem" that "shows an especially strong and clear relationship with the Gilgamesh epic" (p. 402).
If West is correct, and the hero of the Iliad and the main plot of the Odyssey come directly from Gilgamesh, then the Epic of Gilgamesh had a very profound influence on Western civilization: much of Western literature is based to some degree on these two works of "Homer."
Not all scholars posit so specific a relationship, however. A. R. George, in his introduction to the Gilgamesh text, suggests instead that:

Greek poetry imported . . . motifs, episodes, imagery and modes of expression that were always traditional in the narrative poetry of the area . . .[S]hared imagery . . . can be explained as dependence on traditional forms (The Babylonian Gilgamesh Epic, p. 57).

In other words, Homer's work and the Epic of Gilgamesh are parallel, both drawing separately from the same overall set of material that was circulated in written or spoken form around the ancient Mediterranean world. On this view, while the epic itself as a distinct literary work did not directly influence Western literature, the similarities between Gilgamesh and the Iliad and Odyssey reveal that the stories and tales of ancient Mesopotamia provided, as it were, the stones that Homer used to lay the foundation of the Western literary tradition.

The Epic of Gilgamesh is an important historical tool that has attracted the attention of historians and sparked controversy especially due to its perceived contribution on western civilization, or sometimes referred to western culture. The excitement with The Epic of Gilgamesh is largely because of its cultural and religious incorporation, and its impacts on the ancient near east, especially on Mesopotamia in nearly 4000 years ago. Ideally, Gilgamesh is identified as the first ever-great hero with the piece also referred to as the first great masterpiece of modern literature (George, 2002). What can be learned from Epic is the idea of an ancient political structure that included a strong male leader and his council of elders with the rights of citizens also being taken into consideration. This concept of political structure that is traced back to ancient Mesopotamia and revealed by The Epic of Gilgamesh has since had major contributions to how empires and other leadership structures were formed. The beliefs, attributes, religion, and behaviors among others, in the western world, is largely influenced by the stories covered in pieces. Simply put, culture and religion are the central points and extremely influential and relevant aspects of The Epic of Gilgamesh.


Reference
George, A. (Ed.). (2002). The epic of Gilgamesh: the Babylonian epic poem and other texts in Akkadian and Sumerian. Penguin.

It is difficult to determine the exact nature of the influence of The Epic of Gilgamesh on Western civilization. However, it occupies a significant place in the Western literary canon as an early example of an epic narrative. Gilgamesh is the oldest surviving written epic (indeed, the oldest surviving written narrative in general), a form of storytelling which is present throughout most cultures. Even in the present day, we still have epics, in the form of novels, films, and even video games. Gilgamesh's influence spread throughout the ancient world, as there are references to King Gilgamesh by the Greeks and Romans as late as 200 CE. So, Gilgamesh can definitely be seen as a major stepping stone in the history of Western society in regards to the arts.
Gilgamesh also influenced literary studies of biblical books when it was rediscovered in the nineteenth century, since it contains an early variation of the Great Flood tale that is also found in Genesis.
https://webpages.uidaho.edu/engl257/Ancient/epic_of_gilgamesh.htm

The Epic of Gilgamesh is one of the earliest surviving Mesopotamian literary works. It appears to have been used in scribal schools for teaching literacy. It also embodies beliefs about the nature of good rulership that we also see in the law codes of ancient Mesopotamia. 
It is difficult to separate out the influence of the belief system, ethics, politics, and religion of ancient Mesopotamia in general from the influence of the epic in particular. For example, many of the legal codes developed in subsequent empires including the Persian and Roman ones follow some of the patterns we see established in Mesopotamian law codes and much of the legal thought about the duties and legitimacy of rulers echo what is found in Gilgamesh and the ancient law codes. In these cases, though, it might be more proper to say that the epic expresses a system of ideas and values that were influential rather than attributing the influence specifically to one literary work.
Many writers have noted the similarities of elements of the Epic of Gilgamesh and the Bible, including the account of a great flood. Again, one would say that ancient Mesopotamia flood myths may have influenced Judaism, but the epic itself is simply one text recording these myths, and not necessarily a specific influence. 
The nineteenth-century rediscovery of the text of Gilgamesh did have significant influence on Old Testament studies and Homeric studies, causing scholars to increasingly understand the Bible and Homeric poems within a broader historical and geographical context rather than considering them as unique.