I think this is much like the early days of what became the United States.
The US started as a confederacy, not a union. Didn't work out well at all and was subsequently replaced by a much more integrated Union governed by a much more robust set of rules and a more muscular federal role. The Civil War, triggered by cultural and financial tensions, pushed the question of whether states could "check out" of the union to the absolute maximum. Answer? Uh, no. You cannot leave.
What we're seeing in Europe is a variation on that same theme. The EU started as a rather loose set of integrations to avoid massive war (as someone else pointed out earlier). Over the years more rules and regs were added but the EU has a fundamentally weak central legislature that lacks direct election of representatives. So everything really happens through the bureaucracy and regulation rather than true laws, which badly undermines consent and discussion. Sprinkle in a non-mandatory central currency with almost zero central fiscal planning, no common defense, and some pretty different cultural and foreign policy viewpoints and you get something that is unstable.
The financial problems in Greece, the immigration crises, etc. in many way spring from this structure.
If Brexit exposes a practical "you can't leave" aspect of the EU, then things will either bend towards a tighter, true union and/or get very ugly.
I predict the latter first and then the former to follow.
I don't think we'll see cruise missiles flying across the English channel, but we may see the economic equivalent for a time. I hope not tho.