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Salnax said:

Based on military expenditures, North America has the upper hand. According to Wikipedia, the world spent a total about $1.753 trillion in military spending last year. The USA alone spends about $682 billion, over a third of the global total. Canada and Mexico combined spend another 29.5 billion, bringing the total to about $710 billion. The entire European Union COMBINED only spends about $274 billion, and Russia another $91 billion. Even if we factor in random countries not in the EU or Russia, that still only comes out at about 60% of America's military spending.

Of course, this spending has to translate into ships, otherwise there could be no actual shooting each other. Here again though, America has the upper hand. In total, the United States has 10 aircraft carriers. For context, that's more than the rest of the world COMBINED. America also has a majority of the world's cruisers and over a third of the world's destroyers. In terms of sheer tonnage, the USA alone dwarves the combined navies of Europe. Not to mention that Canada's navy is also one of the world's better ones, and that both the US and Canadian Navies are very advanced.

The problem would lie in what happens when US forces reach Europe. Americans talk a tough game, and we even have a lot of tanks and the like. But once North America takes over Britain and establishes a hold on the mainland, then what? North America has about 9,000 tanks. Russia alone has over 22,000. Plus, Ukraine, Germany, Belarus, Greece, and Poland all have quite a few tanks as well. The USA has the most battle-worthy planes in the world by far, but Russia still has over a thousand fighters to use and even smaller air forces like those in West Europe are roughly on par with America in terms of quality per unit.

Most importantly though, we have to remember America's track record. Europe as a continent has about 700,000 people, about 10 times the size of Iraq's and Afghanistan's populations combined. The USA, with a fair amount of foreign help, has barely been able to hold onto these chunks of the world even with some local support. What's more, a lot of American military investment since the Cold War ended has either focused on the Middle East or the Pacific, rather than anywhere close to Europe.

 

So in conclusion, America would be the invaders, and would fuck Europe up, but wouldn't be able to accomplish much of anything.

Agreed, it really comes down to the end game here.  What's the purpose?  Total annihilation?  Colonialism and expansion?  If we're talking total annihilation no troops really need be on the ground.  The US has a vast array of weaponry to reach out and touch people.  Also, the time that Europe has to fortify is important.  Is this a conflict that builds, has warning signs to where it's going, is it a surprise attack?  Who attacked first?  Etc...  All very important here.

If Europe launches a surprise attack, that could push things in one way.  If the US does, it pushes it the other.  If no prisoners are taken, that's another etc...  So many variables.