The power of Rome and the influence of Greece means that modern Europe's culture was shaped by Greece and Rome, and by Christianity.
The Roman Empire covered much of Europe about 2000 years ago, and those places have had a great deal of cultural continuity since then.
Simply incorrect. Get a map of modern Europe and Rome at that time. Then plot not only the Roman territory, but also population and flows of commerce. Compare.
It does not map to modern Europe. It encircles the Mediterranean with much of Europe either not included or on the periphery. Most of Europe was to the Romans barbarian hinterland 2000 years ago. Even at its greatest extent you will see Rome was always centered on the Mediterranean. The "West" would have been entirely alienated from said culture had the flame not been kept alive by the East when the West fell.