Unfortunately, there's no answer to your question because "eastern culture" and "western culture" are not meaningful terms. When you say Eastern are you referring to South East Asia, the Middle East, Indonesia? The "east" is not a monolith and the culture is not universally shared by all of those regions of the world. Even within the Middle East (my region of specialty) the difference between Ottoman Culture and Wahabbi Arabic culture is massive. For this reason, you'd have to be more specific when asking what you are referring to in terms of "eastern culture".

Edward Said argued in his work Orientalism that "eastern" and "western" are terms that exist solely to create a sense of difference between "western" cultures like Europe and the US, and the rest of the world. Said identifies the “East” as the predominant “Other” of European experience, as it is defined in contrast to what are the construed values and principles of “Westerness”. That is to say that the concept of “East” is one that is constructed around preconceived stereotypes and ideas. As a result, it is a tradition of hegemony and domination and fundamentally a power relation, rather than a factual commentary on what the “East” actually is. Orientalism exists as a means of preserving European perceptions of supremacy while allowing “a whole series of possible relationships with the Orient without ever losing him the relative upper hand.” Moreover, Orientalism should be understood as an intellectual and especially so academic process. It is an expression of an attempt to understand and mold the Orient through scholarly practice, given that as Said says “he comes up against the Orient as a European or American first and as an individual second”.

For this reason, in academia the terms "Eastern" and "Western" have no meaningful content. What does it mean to be an Eastern culture? What does it mean to be a Western culture? Is Russia Western? What about Argentina? Brazil? The Philippines? This may illustrate the substantial ambiguity that surrounds these concepts.

I can answer your question somewhat with regards to many cultures in the Middle East. In Europe, after the fall of Rome, many classical era documents were lost or destroyed. Edit: i think I'm wrong about the specifics of this and don't have a source to back this section up. [Most of what we know about this era and about philosophers like Socrates, Plato, and Aristotle comes from documents that were received by the Moors, examined and translated, and then later returned to Europe after the Dark Ages.]

During this time, however, these Middle Eastern cultures studied and learned the teachings of these philosophers. Aristotle was so popular that he was referred to only as "the Philosopher". Early Islamic exegesis and scholarship was heavily based on Aristotelian understandings of the universe. In this sense, the root of "Eastern" culture with regards to the Middle East stem from the very same teachings that allegedly formed the foundation of "Western" culture.

Sources: Following Muhammad, Ernst; Orientalism, Said.

Adding to what was said by Suleiman's Hat the culture of the Classical Age Middle Easterners, the Carthaginians and Ancient Phoenicians appears to have been not much different from the Romans in their strife to imitate the Greeks. Architecture, Sculpture (though notice the "heroic nudity" is veiled) and Representation of their native Gods according to the Interpretatio Greca (that's Melqart/Heracles) is what we now can tell for sure they had in common with the Romans. According to surviving Greek and Roman histories they did read their Greeks as well and Hannibal Barqa was consciously attempting to portray himself as a reincarnation of Heracles. Hasdrubal Clitomachus was a noted Carthaginian philosopher who had outlived his native city because he moved, by that time, to teach in Athens. He has been cited by Cicero and would write works on Roman thinkers, too.

Still the Carthaginians, with their supposed infidelity (the "Punic Faith"), would get a place as the predominant "other", to put it with Suleiman's Hat, as the very opposite to Roman virtue, in later Roman and Greek writers and historians. For that they have been drawing on an earlier tradition of the Greeks depicting the Phoenicians as ruthless and effeminate knaves. Homer has them be slavers. Pindar had them conspire with the Persians to subjugate Greeks (with the Carthaginian invasion of Sicily) and Aristoteles, as Miles believes, was consciously defying these earliest blooms of a Grecian orientalism by praising the Carthaginian constitution in his book on Politics.

Source: Miles, Richard: "Carthage Must Be Destroyed". 2001. Allen Lane.

Now, the Persians did also have a separate literary tradition transmitted partly by the Parsis in India. Or at the very last their scriptures clarify some peculiarities observed about the Persians by the Greeks and certain turns of phrase used by the Persians kings in their surviving inscriptions.

Source: Holland, Tom: "Persian Fire". 2005. Little

So Greeks are not the only tradition there is. These days the popular historian's term for the time they started popping up is "The Axial Age" pioneered by the philosopher Karl Jaspers in 1949, for the time when the classics of philosophy and literature were written in India, in Greece and in China. Never mind it does not work too well if you include, say, Egyptians or Incas, or some even older works from India and China. Still popular ß An arbitrary timespan we're calling "the Axial Age". This term was f.e. dropped by David Graeber while analysing the common pedestrian attitudes towards banking/usury and prostitution in the Axial Age.

The Marxists did think up something very similar while analysing the similar classic scripture attitudes towards women (as property) across the world. They supposed it had something to do with the shift from a Hunter-Gatherer Communism to an Agrarian and Pastoralist society where men could own herds and fields and had to pin down patrilinear relationships. However, you try being a vanilla Marxist in 1949! These days it's not as though; It's mainly matter of taste if you use Marx and Engels or Jaspers and Rogoff to talk about (pre-) history.