I've been interested in this topic for a while and haven't found a decent answer to this.
Obviously there are certain figures, who, during the course of history came to preside over states and empires that dominated vast parts of the European mainland, uniting them under one banner, for example the Roman Empire, and the French Empires under Napoléon. However, it took two (arguably three if you count the Seven Years' War) world wars for the seeds of the European ideal to truly take hold, in the form of the EEC and its predecessors, eventually resulting in the creation of the European Union.
What I'd really like to know though, is at what point the roots for a united Europe started to grow, and the point when general consensus in European countries changed in favour of a union.
In short, what spawned the EEC was World War II.
The EEC was founded on the basis of the Treaty of Paris in 1951, which formed the European Coal and Steel Community.
One of the ideas was to heavily integrate the European nations in order to make us so reliant on trade with one another, we'd never go to war again. It was as much a peace project as it was an economic one, and those ideals have been reaffirmed repeatedly through the evolution of the European Project.
You see, World War II was arguably one of the most brutal, if not the most brutal war ever to take place in Europe. Tens of millions dead, genocide, massacres, widespread destruction on a scale nobody had ever seen before. Everyone was still reeling from the effects of the war.
Nobody wanted a repeat. The EEC was spawned by a supranational surge amongst the populations of Europe. Everyone had experienced World War II and all the horrors associated with it, and nobody wanted a repeat, ever again, especially at the rate these continent-spanning wars were happening at; Keep in mind there was only a 21 year difference between the end of World War I and the start of World War II.
However, this didn't come easily. Charles de Gaulle was avidly against the increased supranationalism, and fought strenuously against any increase of power in the European Parliament or Courts as an infringement on French sovreignty/national interests.
He also vetoed EEC expansion in 1961, which prevented Ireland, the UK, Denmark and Norway from joining. This was primarily due to the UK being viewed by de Gaulle as a "Trojan Horse" for American influence. He was also the cause of the "empty chair" policy the French used in European parliaments and councils, which essentially saw them take a stance of absenteeism from EEC matters.
So, European countries have not always been in favour of a Union and many saw - and still do see - the EEC and its inheritor institutions as nothing more than an economic union, and that was the case up until the Treaty of Maastricht in 1993 which changed the EEC into a political union within the institution of the European Union, and saw new policies adopted such as the Common Foreign and Security Policy(CFSP), as well as the enforcement of "the free movement of goods, capital, people and services".