I think it is more complicated than that.  It may have begun then.  And it was certainly solidified with the lack of European support for the war in Iraq.

But something significant happened in those interim 10+ years.  I don't remember any political animosity toward Europe under Clinton.  Nothing on the radar.  In fact, that was the era of the birth of globalisation and the Internet.  If there was any prevailing sentiment during the 90's it seemed to be one of the world getting smaller and more progressive.

Politically, yes, Europe stood in the way of Bush's policies.  

But there was a cultural change in America that exacerbated any policy differences. I think it was a combination of the "greed is good" mentality of the 90's, the Christian Right's victory during the Clinton Impeachment (he was voted out of office), and the attacks of 9-11, which radically altered the mindset of Americans, made them afraid, disoreinted and extremely defensive.  We became financially corrupt, socially regressive and militarily reactionary while Europe was creating an economic model of cooperation, becoming more socially progressive and enjoying (outside the Balkans) an era of general peace.

You could say we grew apart.  Iraq simply illustrated this.

Those who can make you believe absurdities can make you commit atrocities. -Voltaire