You all talk about "West", but I'd like to see one to define it. I asked once, i'm not gonna push it. So heres my view on the matter.
I dont see the point in defining a broader areal of Western civilization without setting the exact criterias first. Is it religion, government, philosophy, history, arts, sciences, geography and in what way all of these define "Western" legacy? In my mind, for every one of these categories, there are different divisions among the West, so its a little unreal to take Greek culture as the founding core. I'd speculate that this 19th century idea rose because the Greeks were the first civilized people in Europe, and their heritage, (a huge one and amazingly influential) were the first civilization marks native to Europe. But, as a matter of fact, Greeks themselves were part of the Mediterranean cultural complex and shared their identity with other non-European peoples, while the majority of present Europeans were running around wearing animal skins. Greeks exchanged many ideas with Middle Eastern cultures of the time, and certainly Hellenism even pushed this exchange further, to the extent that at the time when Christianity first appeared there was little sense talking about a seperate Greek and other Middle-Eastern cultures. Hellenistic culture consisted not only of, just the opposite, it included the Greek heritage, along with other Middle-Eastern ones.
Furthermore, early medieval Europe was nothing like the Hellenistic world, and a completely different set of values determined its identity, which resembled a dogmatic Judeo-Northern mix. It was only until the Renaissance and the Enlightment that the philosophical basis of Ancient Greece were redescovered and implemented to the degree to really affect people's lives. Not surprisingly, this was a period when religion in Europe slowly began to lose its prime, and imo as a result of that, Western civilization came to be defined as it is today. In this sense, secularism, division of church and state, rationalism and empiricism, rather than dogmatism is what gave the West its core identity, and allowed arts and sciences to freely develop and shape Western identity even further.
During this astonishing civilizational advances in the West, the Middle-Eastern Hellenistic heritage has been shaped by a different ideology, that of Islam. Its relatively stronger grasp over most areas of life (especially in the later period, which lasted till recently) has conditioned the Middle East to take on a paralel path to that of Europe, diverging from the common Hellenistic heritage.
In the end, can one logically ask the question: Is Greece the birthplace of the West? I dont think so. The West is defined as they themselves choose to. The Greek cultural achievements arent Western, they belong to the world. It's up to every person and every nation to choose whether they want to adopt and follow those ideals or not. If West means to embark on such a quest and take it a step further, than so be it. The West stretches from where the sun of freedom sets to the place where the last person is warmed by its rays.