The south didn't vote for Lincoln and attempted to secede with the US president initiating the war to overturn the local state's right to do so.
I think a lot of europe's problems would be fixed by making the office of the EU presidency a directly elected office in a similar fashion to the US presidency. The EU would still require further centralization to maintain the eurozone and to further deepen the EU single market. They are just required to maintain our current levels of wealth and to fix the banking crisis that has never really been solved in the EU.
A single individual with a policy platform that has gained legitimacy through campaigning in all EU member states would settle questions related to the future of the union and will be best placed to implement and direct policies that benefit the majority of the EU population.
If you consider that the US took 4 generations to unify and you consider that present-day it is hard to imagine a peaceful unification then you should also consider that the way you think about things is dramatically different then the way your grandchildrens' children will perceive things.
For example, currently a large part of the EU population speaks english as second language but it's hard to imagine that majority of voting-age EU population to not speak english as second language in 80 years from now.
Speaking a single language will probably clear the biggest hurdle for effective political union.