Ah, Wikipedia.
"Political possibilities
In an interview with the New York Times, teammate George Hincapie hinted at Armstrong possibly running for Governor of Texas after retiring from cycling. In the July 2005 issue of Outside magazine (http://outside.away.com/outside/toc/200507.html), Armstrong himself hinted at possibly running for Governor, although "not in '06." By Texas political standards, he would almost certainly run as a Democrat, as he has described himself in the past as being "left of center," "against mixing up State and Church," "not keen on guns," opposed to the Iraq War, and pro-choice."
I never realized Armstrong was socially liberal, I just thought he was anti-war. Pro-choice, not keen on guns, against mixing up state and church, he'd be attacked viciously if he ever ran for anything in Texas.