Actually it was never de-mobilized because the US needed it to play the role of a world hegemonic power -- to control other places with diplomatic and military might, grab resources and such. It was its opportunity to step on to that role, since WWII had destroyed the European colonian empires.
As for Europe, defend itself from what? With the exception of right wing nuts in power, Europeans didn't feel threatened by the USSR. The people of (non Eastern) Europe, were half and half in favor of communism (huge following in Italy, France, Greece, etc. In Western Germany there were lots of sympathizers too, but the communist party was crashed by Hitler (and then the country was divided post war). That would also be the case in Spain and Portugal, if it wasn't for the dictatorships.
Some Europeans certainly didn't, but many of them, in fact, did feel threatened by the USSR: the ruling classes.
Also, opinions on the USSR were quite divided even on the left, with (Moscow-sponsored) hardcore Communist parties being on very different terms with it than traditional Socialist movements.
[1] https://en.wikipedia.org/wiki/Invasion_of_Canada_(1775)
[2] https://en.wikipedia.org/wiki/War_of_1812
[3] https://en.wikipedia.org/wiki/Fenian_raids
(Does the lack of American effort to impede the Fenian Raids make them an American invasion? It's judgment call.)
The more patriotic among us refer to it as, "The Whitehouse Renovation Project of 1812".
The Barbary threat led directly to the creation of the United States Navy in March 1794 [1]
Then don't forget the Mexican-American war, which started with our annexation of Texas and ended with our "purchase" of 1.2 million square miles of territory at firesale prices from a conquered opponent.
Earlier, Simon Bolivar, who liberated most of Latin America from Spanish control and laid their democratic foundations, loved the American founding fathers, especially Jefferson, and even sent his nephew to the University of Virginia.
Before it could afford that military might, it required a substantial economy to pay for it all.
Land, natural resources, enterprising & hard working immigrants, low regulation, the protections enshrined in the Constitution, ease of starting a business, labor mobility, strong judicial system, checks & balances restraining and slowing down government abuses - just a few of the actual reasons America became what it did.
What makes you think that Carter, who was President during that time and initiated the large military buildup that Reagan is often credited with to maintain that peace, is too naive to recognize that?
"Being confident of our own future, we are now free of that inordinate fear of communism which once led us to embrace any dictator who joined us in that fear. I’m glad that that’s being changed."