The final flicker of the Easter Roman Empire extinguished in 1452 with the fall of Constantinople to the Ottomans. The Holy Roman Empire saw itself as the heir to the Western Roman empire and ancient Greece. And, in a way, we can argue a continuity of vision between the vision of a Europe of Charlemagne and the vision behind the European Economic Community post WWII. However, today's EU has no connection with this vision. As many have pointed out, the monstrosity of the EU is closer to the heirs of the USSR.
So one can argue that the somewhere between 1945 and 2012 the final flicker of the the Western Roman empire went out. The exact date is hard to pinpoint from our immersed perspective but key contributing elements are 1) the culture of self hatred stemming from the holocaust, colonialism, and segregation, that opened the door to the systematic instrumentalisation of the UN Asylumn and Refuge conventions to break the social cohesion of European peoples, 2) for Germany the fatal signing, as a result of pressure from the United States, of the Anwerbeabkommen between West Germany and Turkey in 1961 (for France and GB the ending of Colonialism and the resulting guilt tripping), and 3) the end of the USSR (26 December 1991).
So we could date the final flicker as 1991. The disintegration has been gathering pace since then and of course contributed to the events of 2001, ten years later (see Hamburg cell). The love embrace of left-wing terrorism and Islamic terrorism was the training ground for the Greens like Joschka Fischer and Daniel Cohn-Bendit (1994 he was elected to the European parliament), thinking that became mainstream EU thinking post-1991.