I honestly do not believe that a war against Europe is a winnable scenario for the US. It would be nearly impossible for the US military to defeat the militaries of the UK, France, Belgium, and Germany. And that's not even mentioning if other nations join in and not to mention the factor of partisans conducting guerrilla warfare. Many of the European armies - although smaller and less funded - are extremely professional and do a damn good job at what they do. A large part of the American military can not say the same. Look at the job France is doing in Mali, they've driven the Islamist forces nearly out of the country and it hasn't even been half of a year since they went in. It took the US 10 years to "accomplish" their goals in Afghanistan.
European nations may spend less on their armies but they spend that money in the right places. Into advanced and highly effective firearms and training their soldiers to be professional and highly self-sufficient. There's a reason why American soldiers are now being given calculators... This is not to say that all American military service members are idiots. I have the utmost respect for American special forces, pilots, and the marines because they are self sufficient, highly capable soldiers who know what they are doing. My brother-in-law was in the Marines and fought in Iraq (including the attack on Fallujah) and he clearly remembers several Army soldiers who did not even know how to fire their weapon... It was an m240... a weapon that doesn't even have a safety...