Probably North America, but only because the American armed forces are already unified under 1 banner/general. With Europe split up into different countries it requires someone to take the lead and getting different military branches to work with each other, let alone different countries can have it's issues. Still, it would be ugly, bloody, and both sides would lose tremendously...depending on the rules of engagement.

That supposes that the US would count on Mexico/Canada to hold the line.  And this isn't a slant towards either country, the question is would you leave Canada/Mexico as your only line of defense between a potential multiple pronged land approach into the US?  Probably not. 

with nukes, everything would be destroyed anyway... boom boom boom boom boom... done, all is gone, nobody left to count the casualities and crown a victor...

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.

Bad example. Very much unlike what hollywood movies suggest, WW2 was a war that the US was hardly even involved in. And that is also the reason why they were indeed the big "winners" of WW2 - they waited until all other countries had already pretty much ruined each other before entering the war. That way, they kept their casualties and expenses extremely low.

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.

Russia, China and arabic countries have no love for each other. Why should they form an alliance?

Under normal circumstances they wouldn't, but if I were the Russia, or China, or the Middle East and saw one of these bordering countries falling, I'd have serious concerns about the power shift in the region (Not to mention having the US gain control of your immediate neighbor giving them legal ability to sit there and pile up military resources on a border before doing anything).  Which is a major facet to why the US got involved in Afghanistan circa Cuban Missile Crisis.  If the Russian's took Afghanistan they had a foothold into the rest of the region, not to mention the resource implications (oil, to name one of the most obviously important).  

Anything is better than a Republican.

America has the best miliary and spending by far.

I think i would destroy the world so we all get fucked.

Japan and China would win as when these 2 forces are done with each other they would just split the remains as they see fit.

The problem is that we are not watching the movie "Battle of Midway" anymore. In those 1940ies times, a $1b carrier was essentially invisible to the ennemy and its aircrafts had the "right of first strike". Nowadays, I can sink your entirety of 10 carriers ($10bio) with roughly 200 missiles ($10million). While you try to rebuild your 10 ships ($10billion. 2 years if everybody is in the shipyards), I can take another 100 missiles $10mio, (1 week). In the end, the "power of a navy" has become a relative asset. The question is how do you resupply your troops and how do you distribute your troops if you want to conquer an entire continent?