Not to mention, the Catholic Church didn't start the HRE for about 400 years after the Western Roman Empire fell apart (or 300 years if you consider the HRE to have started with Charlemagne, who really got the ball rolling himself, without the Church). And the connection between the Church and the HRE was somewhat superficial, especially after the Renaissance--it was always under German/Frankish control. While Roman ideas and a few institutions (actually, Catholicism is the only one I can think of) are still with us today, the Roman Empire itself has been defunct for a very long time. Even if you count the Byzantine Empire, which did continue on its own for quite a while, it was a shell of its former self by the High Middle Ages, and was essentially a city-state when the Turks conquered it in 1453. No direct connection to the ancient Roman government has existed since then.