Contador, who secured his second Tour win on Sunday in Paris, said he had no admiration for the American as a person and tensions between the two had a negative effect on the Astana team. Armstrong, the seven-time champion who finished third in his first Tour since 2005, responded that Contador should "drop this drivel."
"My relationship with Lance Armstrong is zero," Contador said late Monday in his hometown of Pinto outside Madrid. "He's a great rider and he did a great Tour. Another thing is on a personal level, where I have never admired him and never will."
Armstrong, who had criticized Contador as being inexperienced earlier this year, responded with his own salvo on Twitter.
"Hey pistolero, there is no 'I' in 'team'. what did I say in March? Lots to learn. Restated," Armstrong wrote.
The "pistolero" remark stems from Contador's habit of celebrating victories by shooting an imaginary pistol.
"Seeing these comments from AC (Alberto Contador). If I were him I'd drop this drivel and start thanking his team. w/o them, he doesn't win," Armstrong added. "A champion is also measured on how much he respect his teammates and opponents."
Contador and Armstrong entered this year's Tour jousting for position as the Astana team leader - an unusual situation in cycling, where team's normally have a clear No. 1 rider who is supported by the rest of the squad.
While Armstrong briefly held a slight edge over the Spaniard in the first week, Contador proved too strong in the mountains and was able to pull away for a decisive edge in the Alps.
On a couple of occasions, Armstrong questioned Contador's tactics during the race, saying they went against the good of the team.
"It was a tense situation," said Contador, who also won the race in 2007. "We didn't have fluid communication despite the fact that we were the two main riders on the team. And this meant the rest of the cyclists and the technical staff also felt a bit of tension."
Armstrong is launching his own U.S.-based team for next year, while Contador's future with Astana remains uncertain.
"Wherever I go I will look for a teammate who is with me 100 percent," Contador said.