Before a live audience at an appearance near Daytona International Speedway, Stewart told Performance Racing Network that he suggested to his Stewart-Haas Racing driver that she challenge the 76-year-old Petty, NASCAR's all-time leader in wins (200) and co-holder of the championships record (7), to a match race.
"I think that (a race) would pretty much settle it once and for all, maybe get him to shut up a little bit, too," Stewart said. "I will supply the cars. If he wants to race her, I'll make sure they have exactly the same setup in the car and give him the chance. He can drive one of my 14 cars. I don't care."
And if Patrick were to win the race, Stewart said, "If I were her, I'd take (the checkered flag) over there and cram it up his (butt)."
Stewart said Petty doesn't understand how much NASCAR has changed since his era, and how much harder it is to be successful now. When Petty drove the circuit, Stewart said that his equipment was almost always superior to that of his competition.
Petty made his original comments about Patrick during a recent appearance at the Canadian Motorsports Expo, stating that, "If she'd have been a male, nobody would ever know if she'd showed up at a racetrack.
"This is a female deal that's driving her. There's nothing wrong with that, because that's good PR for me. More fans come out, people are more interested in it. She has helped to draw attention to the sport, which helps everybody in the sport."
Patrick finished 27th in the Sprint Cup standings during her rookie season, driving the No. 10 Chevrolet for Stewart-Haas Racing. Her highlight came in the season-opening Daytona 500, where she finished eighth after becoming the first woman to win the pole for that event.
That is her only top-10 finish in 46 races at NASCAR's highest level. Before that, she amassed seven top-10 finishes and one pole in 60 races over four seasons in the Nationwide Series.
Patrick has brushed off Petty's criticism, saying he is entitled to his opinion, adding that good things could come from it.
"As I said the last time somebody said something that was not so positive for me, it spawns so many positive articles," Patrick said during Daytona 500 media day. "I love the conversation that it creates in sport, and across the board it makes sports interesting. It makes life interesting when people have different perspectives, and that's fine with me."