The two tried out each other's machinery on a cold and wet afternoon at Watkins Glen, the upstate New York circuit which last hosted a Formula One race in 1980, at an event organised by a mutual sponsor.
"It's just amazing what the capabilities of the car are," SpeedTV quoted Stewart as saying.
"I told the guys on pit road out there that it's probably going to make my (NASCAR) crew chief a little more stressed during the weekends because I'm going to want it to handle like that all the time.
"I don't think we're ever going to get it there."
Hamilton, who left Montreal with the likes of former champion Niki Lauda criticizing his driving as "beyond all boundaries" and "completely mad," enjoyed a far more enjoyable afternoon than he had on Sunday.
"Pretty cool. Probably the coolest thing I've done outside of racing Formula One," the Briton said after doing a final burnout for some 10,000 fans.
"I was also feeling the tough weekend this morning but as the excitement built up, and when I got in the car, and once I got out, I completely forgot about last weekend."
The 26-year-old Hamilton has had a rollercoaster season, winning in China but repeatedly called to see the stewards after collisions and controversies. He has crashed out of his last two races.
His Montreal clash with Button, which put Hamilton out of the grand prix while his team mate went on to win, was a major talking point after the race -- as was a reported meeting in full view of the paddock with Red Bull boss Christian Horner after Saturday qualifying.
That meeting led to some speculation about a possible move by Hamilton to the champions and a denial by the Briton's agent that the driver was in discussion with any rival team.
(Writing by Alan Baldwin in London, editing by Tony Jimenez)