If current events in Europe have shown us anything it's that the West isn't nearly the collective entity as people believe it to be. The standard definition of the West as embodied by individualism, constitutionalism, equality, rule of law, separation of Church and state, human rights, free markets, democracy etc. are actually Anglo-American ideals, not pan European.
No, these ideals are indeed pan-Western European. All the ideals that you mentioned developed in the larger context of the 18th century Enlightenment, which was a pan-Western European phenomenon. The Enlightenment, along with the earlier phenomena of the Renaissance and Reformation (and Counter-Reformation) played a major role in shaping the modern Western civilization. There were numerous prominent intellectuals of the Enlightenment who were not of an Anglo-American background, including Montesquieu (a Frenchman), Jean-Jacques Rousseau (Swiss), Immanuel Kant (a German) and many others. These figures and others contributed greatly to the civilizational fabric of the West - including the Anglosphere - and made it what is today. The concept of separation of powers between the legislature, judiciary, and executive, for example, was conceived by Montesquieu, and was later adopted by the American colonists following their revolution. Throughout Europe in the late 18th and 19th centuries, revolutionary movements such as the French Revolution that demanded republicanism, secularism, constitutionalism, and other such ideals that we associate with the "West" were largely indigenous phenomena that were inspired by broad Enlightenment values rather than specific Anglo-American ones. The French Declaration of the Rights of Man in 1789, for example, was a unique and revolutionary document that had no real counterpart in Britain - indeed, the British regarded the republican movement in France as one of dangerous radicals and mobilized considerable resources in the aid of pro-monarchy reactionary forces.
Here too it's an example of the victors writing the history as they see fit. I'm glad you brought up Sam Huntington, because his entire outlook was shaped by the ideological divide of the cold war where the "West" as he saw it was pitted against the rest. Except, the West he kept on referring to was the post War Europe which was overwhelmingly dominated by the United States and England. Anyone who has traveled/lived in Europe can attest to the fact that when it comes to ethos, the Italians or the Spaniards have virtually nothing in common with the likes of the Germans or the Swiss or the Scandinavians.
The Italians and Spaniards have a whole lot in common with the Germans and other North Europeans. They share a common Judeo-Christian tradition, which is common throughout much of Europe and historically played a huge role in shaping Western civilization, and share a similar worldview and agree on much of the same basic ideals that were mentioned above. Their cultures and nations were both shaped by the same historical processes that were common to all of Western Europe. The relationship between a Spaniard and a Scandinavian is somewhat similar to the relationship between peoples from two different parts of India - say, a Telugu and a Punjabi. There are great differences between the two, to be sure, but there are also great commonalities, and these over-arching commonalities are what constitute a common civilization.
I do not agree with many aspects of Huntington's thesis, but I do agree with the general notion that the world can be divided into broad, cultural categories called "civilizations". No one is claiming that these civilizations (including the West) are monolithic, homogeneous entities - they are far from that. But neither can anyone deny that the various nation-states of what we call the West have much more in common with each other than they do with the nations of any other place in the world, owing to their shared history and culture.
For all the greatness of the Greco-Roman ideals, neither Greece nor Italy would survive if it weren't for other wealthier states propping them up. Heck some people don't even bother including Greece in the West. The fact of the matter is that the world is a far more complex place than what most people both in the "East" and the "West" believe it to be.
Modern Western civilization is NOT synonymous with Greco-Roman civilization, which is extinct. Although some aspects of Greco-Roman civilization were literally resurrected during the Renaissance and had considerable influence on the subsequent development of Western civilization, the two entities are quite distinct from one another and should be differentiated. The reason why Greece is not included as part of the modern West is because Greece did not undergo the same historical processes like the Renaissance, Reformation, and Enlightenment that Western and Northern Europe did, and developed in a different manner owing to their long occupation by the Ottoman Turks and relative isolation from the rest of Europe.