European reluctance to send troops to Iraq or Afghanistan is more than just the result of political differences. While Europe has about twice as many troops as the United States, they have far fewer fit enough to ship to a combat zone. This was a problem first noted in the 1990s, when there was a big demand for peacekeepers in the Balkans. The Europeans couldn't fob this one off on the Americans, and had to come up with combat ready troops. The Europeans had a tough time finding soldiers ready and able to go.
European armed forces are full of people in uniform, who have a civil service mentality. That is, they think, and act, like civilians, not soldiers. Belgium recently discovered that 14 percent of its troops were obese (compared to 12 percent of the general population), and unfit for many of their duties. Much noise is being made about getting all the troops in good physical shape. While that is possible, it is less likely that the mentality of the troops will be changed.
During the Cold War, Europe got most of its troops via conscription. Young guys came in for two or three years, and then left. Anywhere from a third to half the troops were long term professionals, in for twenty or more years. But even before the Cold War ended, many of the European military professionals were losing their combat edge. When the Soviet Union disappeared in 1991, there was no longer any compelling for a European soldier to think and act like one. It was just a job. A government job that was not, or should not, be terribly demanding.
The Europeans spend a much higher proportion of their defense dollars on payroll, leaving little money for training, new equipment and maintenance. It also means an older, on average, bunch of troops. Going to war is a young man's game, but the Europeans have instead turned their armed forces into another job creation program. Britain is one exception, demanding that the troops remain fit, and maintaining high training standards. Most European nations maintain a few elite infantry units, but these don't add up to much in terms of numbers. Only Britain and France have large "rapid reaction" forces that can be sent overseas on short notice. The United States has the largest such force, and many European nations are trying to expand theirs.
The U.S. has long maintained an "up or out" promotion policy, which forces people out of the service if they are not promoted within a certain amount of time. By also maintaining high standards for new recruits, and career soldiers, America is able to maintain more combat capable units. The U.S. is able to field more combat troops, and far more combat power, than over twice as many European soldiers, sailors and airmen.