Another problem with the scenario listed is that in such a situation, it wouldn’t be one-on-one. A nation’s allies are extremely important in war, and people would be lining up to pick sides. We have to take that into account.
On our side of the Atlantic, we would definitely have Mexico as our allies. Canada is iffy…though there is a good chance they would consider allying with the EU, the populace probably would not get behind it, it could definitely lead to Canada splitting up. 90% of Canada’s population lives within 100 miles of the longest undefended border in the world, and we outnumber them 10 to 1. Unless the EU has been building bases and stationing armies in Canada prior to this hypothetical war, Canada is not a threat.
South America will be divided probably, some countries favoring one side or the other, some neutral, it would even itself out and have no major influence on the actual conflict.
The most of the Middle East is probably going to ally with Europe, Israel with the U.S. This could be a problem if we divert forces to defend Israel, is not that important if we are only looking out for ourselves in the conflict.
I think Russia is going to side with the U.S. They did in the last one and got a lot out of it, they will sacrifice a few million of our guys to avoid getting into it with us and to have a say in how Europe is carved up after we win. Japan will as well. China will probably remain neutral (especially since there is still animosity towards the Russians), though they might side with us more openly if Russia doesn’t side with us, or if some of their other, weaker neighbors side against us, hoping to occupy our enemies land during the war and hopefully hang on to it otherwise - a lot depends on who Indochina and India side with, which is a toss-up in my opinion.
I think the U.S. will win. If we don’t trash Canada after conquering it and if the EU has pressure on the Eastern front, the UK will probably drop out of the EU and side with us. If we are to invade we will probably be coming across the Channel. I think the French would cave, not so much because of a somewhat ill-deserved reputation as cowards, but because they know that we would bomb the shit out of them before attempting another D-Day. With the UK our allies, France occupied, and Russians and possibly Americans hitting from the east, they would crumble.