Merely connecting people won't act as a barrier to war, but I think forming a universal governing structure would; however unlikely it is to actually happen. There's a saying that states democracy is just a farce to control the masses by giving the illusion of control, well I would say the EU does the exact same thing.
I can't speak for EU politics of late, I've rarely been in the UK the past year, however last I was keeping track the UK had a major problem with French farming subsidies (basically French farmers were being paid to be uncompetative) and the French wanted the UK to stop getting discounted for payments to the EU (each country pays like a tax that's supposed to benefit the whole, however the UK was paying less than it was getting in even though we're the richest country in the EU).
So, personally, I would say no level of communication will prevent war unless you're economically and socially tied as in the EU. I mean Kings would marry their daughters to Kings of other countries in an attempt to prevent war, the British bloodline has French descent in it. Even look at Richard the Lionheart, he barely even spoke english! Yet no amount of ties between England and France ever stopped war until the EU. Simple fact.