Like, when did they stop worshiping them and try to recreate them (or a new version of them)?
Many would argue I think that Europe never has 'moved on' from idolizing the Roman Empire. The European Union could be seen as an entity attempting to create a unified 'empire' - similar to Diocletian or Constantine's policies that involved religious persecution or a 'Gold-standard' in order to create unity throughout the Empire.
That being said, I believe that 'moved on' would include a certain type of rejection or dismissal of the past. And under these criteria I don't believe that Europe - or much of the Western world in that matter - has ever 'move on' from idolizing Rome. As I'm sure you know and have heard, much of todays foundations in law and economy was developed in Rome or Byzantium under Justinian and Theodosius (a fair argument can be made I believe for the classification of Byzantium not as a 'new Rome' but rather still the Roman Empire)