The Roman Empire is perhaps, the genesis of western culture. While the Greek city states had a part to play, as did the Roman Republic, I believe the Empire was instrumental in spreading that culture across a divided continent. The language of Latin has widely been forgotten and is no longer spoken and the Romans themselves have ceased to exist, the traditions they fostered lasted for over a millenia. The Roman Catholic Church is perhaps the greatest contribution to the Western world. As the Catholic Church maintained a stranglehold on Europe long after the Roman Civilization itself ceased to exist.
But, the Church is not the only contribution to the Western world. The Roman Empire fostered a tradition of democracy, even during the Imperial state. The senate remained a powerful body of politics, intrigue, and power. The Roman Civilization spread demoracy to the far corners of Europe and while those traditions might have been fogotten in the Dark Ages, they would reapper far in the future. Revolutions such as the American Revolution and the French Revolution would use Roman teachings on demoracy and republicanism to found new nations based on ancient Rome. The capitol of the United States is entirely based on Rome, from the white marble, to the pillars. The founders of the United States even used false names in newspapers when they argued for revolution and the constiution. They adopted aliases such as Ciecero.
Other contributions widely used in the Western World besides christianity and democracy is roads. The Roman state was the first to create modern highways and road systems. Many of these roads still exist in Europe and in North Africa. The Romans used roads, made advances in mathmatics and science and geography. They plotted out and mapped out all of their provinces and along with it most of Europe. While, they eventually subcummed to the Germania states, their teachings and philosophies were not forgotten. The Church would survive and while the fire of democracy was snuffed out it would eventually resurfice to dispose of the Anglo-Saxon kings that took the place of Emperors.
In my opinion the Roman Empire was instrumental in spreading the ideals of Rome across Europe. They created the modern ideals of republicanism and the Church that would guide Europeans until the Reformation.