There are lots of things that the supporters of the Euro, and the Common Market, and the broader project of which both are a part, hope to get from increased European unification. But the one big thing is future peace. The first half of the 20th century featured two horrific wars, originating in and largely fought in Europe, largely between European states. Behind all the talk about the convenience of a common currency or the advantages of free trade within the EU is that memory, and a burning desire that it not happen again.
Which raises two interesting questions, to neither of which I can offer a confident answer:
1. If the EU dissolves, with countries going back to separate currencies and separate trade policies, is there any significant risk of a third major European war within, say, the next fifty years? My gut reaction is that there is not, but I do not have any real support for it.
2. If the EU is maintained and European integration increased, perhaps along the lines that the supporters of the Euro have been urging as necessary to save Greece, Italy, Ireland, and Spain, is there any significant risk of a third major European war within, say, the next fifty years?
Before dismissing the possibility out of hand, it is worth considering what came out of the "integration" of the American states. I am not sure how much of a stretch it is to imagine a future where some of the countries, such as Germany, feel that they are being outvoted and exploited by others, such as Greece and Italy, through the institutions of a United States of Europe, with the tension eventually exploding into civil war. Each side would, of course, start out confident that once it was clear they were willing to fight the other would back down.
(Might make an interesting sf plot.)
Which suggests a third question: Does European integration make a major European war less likely, or more?