It depends on who you ask. Personally I see the Egyptians as the creators of western civilization with the Romans perfecting it. Many of my professors and fellow students would not agree though and here are a few arguments that they might have:
Democracy was invented in ancient Greece, which is often considered to be a trait of a truly civilized society.
Laws were first written down and viewed to the public in ancient Greece. Laws wasn't necessarily first in ancient Greece, but the fact that anyone could see them was.
Philosophy was invented here. Socrates, Platon and Aristoteles are widely known personalities and this philosophy was a stepping stone towards science.
Comedy and tragedy, along with theatres, were first seen here. The ancient Egyptians had plays, but no arenas specifically made for them.
Although Cicero is usually viewed as the father of rhetoric, it was actually sort of invented in ancient Greece through Demosthenes.
Well on the one hand it is pretty hard to call Greece the birthplace of civilization in the west. Agriculture, a settled form of life in villages or even cities, writing, complex forms of administration, written law and many things more were all developed in the Fertile Crescent and came from there to Greece. Ancient Greece owes much to its neighbours to the east. For example a period of ancient Greek art history, roughly from the late 8th to the 7th century B.C., is called the "Orientalizing phase" by archaeologists because of the huge impact that art from Syria, the Phoenicians or Egypt had in Greece around that time.
On the other hand ancient Greek civilization itself was certainly highly influential in Europe, the Mediterranean and the Near East and at least from the Hellenistic period onward it shaped pretty much every other culture it came into contact with in some way. That especially applies to Rome, which was so highly Hellenized that you can't study it today without also dealing with Greece. Even aspects that today are regarded as "typically Roman" like law or portrait sculpture have greek antecedents.
The other pillar of western civilization that is often sited besides the Graeco-Roman culture is Christianity or Judeo-Christian culture. And here also you can't really get around Greece. Talmudic Judaism and early Christianity had their formative years mostly under the Roman Empire in its mostly Greek eastern parts. Christianity would never have evolved the way it has without the influence of Greek thought.
But what exactly is "western civilization"? A combination of Graeco-Roman culture and Abrahamic religion? Certainly. But that also applies to the Islamic world, a culture that today often seems as the very antithesis of the west. Also, like stated above, many other factors besides Greece, Rome or ancient Palestine had their share of influence on the development of the west.
In the first place, I think, western civilization, like many forms of identity, is a cultural construct. At least from the Renaissance onwards people in Europe and later also in America looked back at ancient Greece and felt a connection. They emulated ancient art and architecture, build on the knowledge and achievements of Greek philosophers and generally shaped their own culture on ancient templates. In a way by thinking of ancient Greece as the birthplace of their own culture they themselves made it true. And that is still true today, although our culture is always shifting and transforming. Ancient Rome or Athens still loom larger in the public consciousness than Carthage or Persepolis. In 300 the Greeks are the good-guys you're supposed to identify with, not the Persians. In schools you can learn Latin and Ancient Greek but not Punic or Aramaic.
So in the end I would say yes, ancient Greece is the birthplace of western culture. Not necessarily the historical ancient Greece, although that had a huge impact too, but the one in the imagination of the people in modern western societies.