I have to say it would depend where the war was, what the limits are and who else is involved. Let's face it, any war with either Russia or the US involved would pull in alot more players. The US would have most if not all NATO countries, possibly alot of EU countries too. Russia would probably have China and some of it's most loyal satellite states. A war in Europe would probably be a pretty hard war to even draw for the US side, the sizes of the forces arrayed against them are enormous. After all, most of the weapons Russia has or had are still in storage, hell if they ran out of T-80s and T-72's, they still have a thick pillow of tanks to sit on. I think they still have a couple of hundred T-34/85's, not sure if that is true, but still that means there is alot of armour sitting around for their use. Sure the US side have superior tanks like the Abrams, Challenger II, Leclerc and that, however once they have suffered some damage from the constant waves of Russian steel, there isn't much left. I am pretty sure that nobody else keeps their old tanks for a special occasion, the US flogged a bunch of their M60s to Israel and I am not sure what the hell we Brits have done with our Chieftains. Just on the tank front, the ex-Soviets seem to have us out.
Onto aircraft, most of our aircraft are superior to most of their aircraft, however the Russians and their satellite state's have huge amounts of MiG's of different versions and capabilities. There are something like 100,000 in total of all AIM Sidewinder's, not alot of those are that modern, so quite afew will miss. Missile supply is one of the main problems for the US side to contend with on the air front, it would only be so long before the waves of enemy Fighters and Ground Attack aircraft have their wicked way with the US side.
Infantry wise, there probably isn't much of a difference between the average Russian and European soldier's training and equipment, only the British and Americans really have an edge in training and probably only the US Army has an advantage in weaponry.
All in all, there really isn't much to support the US side winning without massive losses to say the least. Feel free to pick apart my argument, I am only remembering facts not trying to find sources.