Very interesting discussion.
First, to answer Fire Haired's question, culture is never dependent on ethnicity. Culture is by definition acquired, and it is partly acquired through language. Therefore any American, whatever their ethnic background is culturally American.
Western culture is derived from Western civilisation. The term 'Western' is a modern invention aimed at bringing together the cultures of European countries and their former colonies and to oppose them to other "Eastern" (from a Eurasian point of view) civilisations, be it in the Middle East, the Indian subcontinent or East Asia. The common issue with the definition of Western is where to place the geographic limit to that Western part of Eurasia. Should it include Russia, which is as eastern as the Middle East, and from the 18th century even expanded all the way to East Asia? In ancient and medieval times, the "Western world", for Europeans, also referred to the Western Roman empire, the Catholic sphere of influence, while the "Eastern world" referred to the Byzantine, the also Orthodox spheres of influence. This neat division between western and eastern Europe took on a renewed significance during the Cold War, when Europe was once again divided almost perfectly in the middle, but this time with Latin and Germanic countries in the western capitalist half, and mostly Slavic countries in the eastern communist half. It is these two historical divisions within Europe that are usually problematic in the definition of 'Western' culture or civilisation. Should it be Western as in Europe vs Asia, or Western as in Western Europe vs Eastern Europe? Both meanings have been used, but since the end of the cold war, the spread of capitalism (and of the EU) to former Eastern Bloc nations, Westernness now tends to extend to all Europe. Needless to say that politics plays a major role in how we decide to define Westernness.
So what is Westernness and when did it really emerge?
Some definitions include the Christian religion, or at least a Christian heritage, but if that were to be a defining characteristic, then Christian Egyptians (Copts), Syrians, Lebanese, Armenians, Indians or even Koreans could also have some claim to Westernness, which in my opinion they don't.
Language is an essential part of any culture, and indeed people almost always tend to feel culturally closer to people who speak the same language as them (see the English-speaking world, Spanish-speaking world, Arab world, etc.), or a closely related language (Germanics between them, Romance speakers between them, Chinese people whose "dialects" differ as much as Romance languages). One thing that helped shape the modern concept of Western culture is actually the English language, which is a hybrid of Germanic and Romance languages. After becoming the most widely spoken European language, and the most important politically, scientifically and so on, speakers of Germanic and Romance languages started feeling a bond between each others via English, a language that is easy to learn for either group. Many people underestimate (or completely fail to consider) the unifying influence of the English language on the Western world. Now English has become a truly international language in its reach, but it is also the language most strongly associated with modern Western culture or civilisation (whether that pleases other Europeans or not).
All the Indo-European linguistic family is ultimately related, so if language relatedness alone was the key to Westernness, North Indians, Iranians and Kurds would also be Western. That is obviously not the case since the term Western was always meant in geographic opposition between Europe (Western or all) and Asia. In any case, Indo-Iranian languages split 5000 years ago from the rest of IE languages, a rather long time to claim a close cultural tie. In fact, civilisations did not yet exist back then, so it would be illogical to claim that all Indo-European belong to the same civilisation.
Historians usually talk of Babylonian, Assyrian, Phoenician, Egyptian, Greek, and Roman civilisations, clearly distinguishing each entity, even if they were relatively close to their neighbour or predecessor. Yet we talk of a single European or Western civilisation (don't be fooled by Sid Meier's Civilization game series, there is no such thing as French, German or English civilisations).
In my opinion, Western civilisation and culture really emerged in the Renaissance. Europeans did inherit a lot from the ancient Greek and Romans, but so did the Arabs and the Ottomans. In fact, ancient Greek culture survived Roman rule, and Greek, not Latin, was the official language of the Byzantine Empire for over 1000 years. Before the Romans he Greeks/Macedonians under Alexander had conquered all the Middle East. So for most of ancient and medieval history, from 330 BCE to 1453 CE, the Eastern Mediterranean and a variable portion of West Asia was under Greek rule or Greek-language administration. The Greeks undeniably have a stronger historical presence in the Middle East than in the rest of Europe. Even after the Ottomans took over, Greece remained firmly inside the eastern Middle Eastern civilisation.
I do not mean that modern Greeks, since they decided to split from the Ottoman Empire in 1821-32 have not since become more culturally European. In fact they were always European, but not 'Western' as their empire was an Eastern one (the terms Near East and Middle East match the borders of the Macedonian and Byzantine Empires). Therefore by definition ancient Greece and the Byzantine Empire cannot be considered Western. In fact, there was no concept of Westernness or Europeanness in ancient times. Germanics didn't feel like they shared anything in common with the Romans and vice versa. The Roman Empire was European, Middle Eastern and North African, not just European, and certainly not pan-European.
It is only during the Middle Ages that Europeans started to feel a sense of shared identity. Christianity played a unifying role in that regard. That process really started with the Muslim conquest of Iberia and the Viking invasions, two events that forced the Franks and their allies (English, Irish, North Spanish kingdoms) to stick together against their common ennemies. The Vikings originally felt completely separate from Christian Europe, but this changed afer they converted. The crusades, then the Ottoman conquest and their advances into the Balkans also reinforced the sense of unity among Christian Europeans. However, Western culture did not yet exist back then. The feeling was barely nascent. It would require the develomment of the set of values and ideals that were going to emerge from the Renaissance onward. That included rationalism, but in my opinion not democracy or capitalism, which are latter additions that didn't take root in all Europe until the second half of the 20th century. In fact, it could be argued that democracy is just an illusion, and that the Chinese were capitalist-minded long before the Europeans.
The constant intermarriages between European monarchies and the establishment of pan-European empires like that of the Habsburgs, which spanned across half of Europe, from Spain to Poland and from the Netherlands to Sicily, gave Europeans a new perspective on shared Europeanness. Even the King of England once owned nearly half of France, before unifying England with Scotland, and for a time with the Netherlands and Hannover. All European kingdoms became interlinked. More than anything else it is the emergence of a common culture among the European ruling class, essentially the aristocracy, with a lingua franca of their own (French from the Renaissance until the early 20th century, then English) that laid the grounds for Western culture. Culture, like language, is typically imposed by the elite on the rest of the population. Modern European languages have their roots in the dialects spoken by the ruling elite in each country during the Renaissance. Standard French language, for instance, is the Parisian dialect as psoken by the royal court that was eventually imposed on the whole country (although not until the advent of universal education and mass media). Modern Italian was modelled on the Florentine dialect, which was considered the most prestigious. Standard English evolved from the London dialect of the Renaissance.
So Western culture emerged during the Renaissance among the European elites and matured over the next few centuries, still mostly among the aristocracy and the bourgeoise. But it did not reach all the strata of the population until the 20th century, especially in remote rural areas. That is why people like the Amish, who cut themselves off from the rest of society in the 18th century, are only partially Western. On the other hand, a number of countries that were not traditionally Western, such as Japan, Korea, the Philippines, or to a lower extent Turkey, are becoming increasingly Westernised. Since culture can be acquired and is not linked to ethnicity, it is technically possible for any country to adopt Western ways and values. Japan was the first country in history to intentionally decided to replace a large part of its own culture and system by the Western one, including for clothing, political and economic systems, the adoption of western science and rationalism, etc. I summarised those changes
here.
