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If you placed the two armies ready on a battleground, all other things being equal, the European army should win. That is one of the reasons why some EU countries, notably France and Germany, are not wholly enthusiastic about NATO and the special influence the US has in that organisation.
But in a (non-nuclear war) presumably ranging back and forth between the US and Europe, the US would have an advantage. Few European countries have forces that are intended to fight (far) outside their own territories, whereas the US military is planned largely on the assumption that it will be fighting overseas. Going up against Europe, the US would be handicapped in that many of its jump-off points are in Europe, but still it should be better placed to attack Europe than vice versa.
Of course if the conflict moved from bombing to invasion, then a lot would depend on the reaction of the population. That's largely an unknown as Americans haven't fought at home since the Civil War and few Europeans of combat age have personal experience of war either. I tend towards believing Europeans would bear up better, but really, who knows...
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