Tipsarevic drew Serbia level at 1-1 against the Czechs in their quarterfinal, but he was pulled away by his captain from a confrontation with Stepanek following the handshake at the net.
“After five hours and 10 minutes, instead of shaking my hand, he gave me a middle finger, and told me that I'm a stinking, something bad...” Tipsarevic said.
“Unbelievable. I never ever thought that I would experience this. It's pathetic, nothing else.”
Images appeared to show the two players shaking hands, but Tipsarevic was visibly angered. Stepanek afterward denied any conflict.
“We normally shook hands,” he said. “He was likely happy that he won.”
Czech captain Jaroslav Navratil said that “Radek may have told him something, but I was too far from them (to hear).”
Tipsarevic, who this week reached his highest ranking of No. 8, faced a match point at 5-6 and two more at 6-7 in the fifth set but served well to remain in the match and prevail in more than five hours.
In a thrilling final set, Tipsarevic broke to lead 8-7 and served the match out, converting his first match point with a backhand winner.
Earlier, Tomas Berdych eased past Viktor Troicki 6-2, 6-1, 6-2 to give the Czech Republic a 1-0 lead after the opening singles on an indoor clay court at the O2 Arena.
Trailing 3-1 in the first set, Stepanek fought back, breaking Tipsarevic twice. The Serb then went 4-2 up in the second set before serving it out.
Tipsarevic again broke Stepanek in the opening game of the third set and served it out with a backhand winner. Looking in control, he took a 4-2 lead in the fourth set but the Czech managed to come back, winning four consecutive games and forcing the fifth set to cheer the sellout partisan crowd of almost 14,000.
Czech pair Frantisek Cermak and Lukas Rosol have been selected to play Ilija Bozoljac and Nenad Zimonjic in Saturday's doubles, but captain Jaroslav Navratil is expected to bring in Berdych and Stepanek who are 9-1 in the Davis Cup doubles. Changes are also possible for the Serbian pair.
“There's no other option than me and Tomas,” said Stepanek. Tipsarevic said he was not sure yet if he would play with Zimonjic.
The seventh-ranked Berdych never lost his serve and converted his first match point with a return winner to complete the victory in 1 hour, 49 minutes.
“I did what I could,” Berdych said. “When you get prepared well, anything's possible.”
Troicki opened the match with a double fault and struggled throughout on a slow indoor clay-court at O2 Arena and lost his serve seven times.
“Obviously, I played bad from the start,” Troicki said. “I was too defensive and he gave me no chance to come back. I was really disappointed.”
The Czechs played their recent home Davis Cup ties on indoor hard courts but hoped the clay court would work in their favor.
In the opening singles, the change worked.
“The court suits us well.” Berdych said. “That's exactly what we wanted.”
Berdych dominated his 27th-ranked opponent despite 30 unforced errors to improve to 18-11 in the Davis Cup singles and 3-1 against Troicki.
“The most important (thing) was that I could rely on my serve,” he said.
Tipsarevic already helped Serbia overcome the Czechs in their last Davis Cup meeting in Belgrade in 2010, beating both Berdych and Stepanek as Serbia won 3-2 en route to the country's first title. He is 32-12 in Davis Cup singles.
Serbia, seeking to reach the third consecutive semifinals, is missing top-ranked Novak Djokovic, who opted to take a rest before he opens his clay court season at the Monte Carlo Masters on April 14.
The winner of the best-of-five series will face Argentina or Croatia in September's semis.