UK votes to leave EU

I think the cultural roots go to Greece, not Rome. Rome basically was a logistical and administrative juggernaut but I'm not aware of anything new they created - Roman law being the only exception, I think, which created basis for the laws in a large group of european polities.

After Rome, there was the period which is called the dark ages, but probably mostly only looks like that in the context of the exceptional developments which started after it.

1400:s europe finally was done with dark ages, and the energetic cities embraced innovations from around the world (including material from greece antiquity preserved in the muslim civilization).

The catholic church did not really do much to help that progress. You will notice that e.g. industrialization was fastest in the countries where the church did not have as much power - the mostly protestant ones. Big parts of what we call 'western civilization' was innovated ny guys despite the stifling effect of the church on sapient inquiry.

In the border regions where a bunch of tiny polities that imitated and admired the developments of biggest neighbours.

My point is, just saying 'rome and christian culture' is giving undue prominence just to a one cultural development that moulded europe.

The shared dream did, indeed, mean exterminating the current residents. The current residents were often described in news accounts from the time as inhuman savages, not unlike how Africans were viewed. While it's not something I'd want to put forward as key to our identity, it most certainly was key to our identity. Predating the Manifest Destiny, the people of what the United States saw themselves as the right, God-given rulers of the land. It's disgusting, and lead to one of the few "successful" genocides in human history. But it most certainly did unite the early United States of America.

Shared culture? Yes, there were many different religions in the early United States. However, they all had a united cause--they were fleeing religious persecution in their home country. This includes the Catholics (fleeing England), Lutherans, Calvinists, Quakers, and Unitarians. It is so much a part of US history that the US Constitution guarantees religious freedom as its first Article. Religious freedom is at the very heart of the founding of the United States, and it is a result of this realization that many American immigrants were fleeing religious persecution in their country of origin.

So, basically, the differences between the US and the EU are exactly as the GP mentioned. They were very strong allies to France--the US constitution even includes almost direct quotes from French writers of the time; France both financed the US war of independence and the US bailed out France from the Napoleanic wars by purchasing the Louisiana Purchase. And while they didn't share the same religion, many fled from religious persecution--it was such a common cause that it was enshrined as the first Article to the US Constitution. The US was, indeed, actively engaged in genocide. This, at least, is one thing the US shared with Europe. Europe, from at least the time of the Crusades, but continuing through the slave trade and the many genocides in the 1800s and 1900s, most certainly shares this thread in common with the United States. Congratulations on making the point so clearly that the two cases, in this fashion, were very similar.

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