The first precursor of the EU was the European Coal and Steel Community [1]. It had the express purpose of putting control over coal and steel under supranational control, thereby making war between the members materially impossible.

It's also important to note that NATO does not provide prosperity (that was the goal of the EEC through closer trade and cooperation). So it does absolutely nothing to prevent the rise of extremist movements that could cause a repeat of the 1930s. There is also nothing it can do to prevent a member from quitting, just like Germany quit the League of Nations.

[1] https://en.wikipedia.org/wiki/European_Coal_and_Steel_Commun...

The economic structure of the EU provides incentives to avoid conflict.

Worth noting that two NATO members (Greece and Turkey) have had deaths from a not-quite shoot out http://edition.cnn.com/2006/WORLD/europe/05/23/greece.crash/. Interesting also that Turkey isn't an EU member, and counties like Germany and France have had as least as much a history of conflict as Greece/Turkey have.