Michael Messerly, a fan who witnessed the crash, said it appeared Stewart struck the driver as he tried to speed past him.
"The next thing I could see, I didn't see [the other driver] anymore," Messerly said. "It just seemed like he was suddenly gone."
The accident came just four days after the one-year anniversary of an accident in a sprint car race in Iowa in which Stewart suffered a compound fracture to his right leg. The injury cost him the second half of the NASCAR season.
The track, about 30 miles southeast of Rochester, canceled the remainder of Saturday's race within five minutes and later posted a message on its Facebook page encouraging fans to "pray for the entire racing community of fans, drivers, and families."
It said a statement on the crash would come later Sunday.
NASCAR driver Dale Earnhardt Jr., whose father died as a result of an on-track accident, tweeted about the tragedy Sunday morning.
We will all loose someone in our time. When a loss is sudden and unexpected, the pain & sadness is suffocating. Prayers for the Ward family.- Dale Earnhardt Jr. (@DaleJr) August 10, 2014
Ward's website said he began racing go-karts in 1998 at age 4, but didn't start driving sprint cars until 2010. He was Empire Super Sprint rookie of the year in 2012 and this year was his fifth season racing in the Empire Super Sprints.
Stewart was involved in a July 2013 accident at Canandaigua Motorsports Park that seriously injured a 19-year-old driver. He later took responsibility for his car making contact with another and triggering the 15-car accident that left Alysha Ruggles with a compression fracture in her back.
On Saturday, ambulances, fire trucks and police arrived within minutes of the crash, Messerly said. Fans filed out in stunned silence, he said.
Stewart only returned to sprint track racing last month, almost a year after breaking his leg in the crash in Iowa. He didn't return to racing in any form until February, when preparation for NASCAR's season-opening Daytona 500 began.
He refused to stop his extracurricular racing despite the injury. The multimillionaire is known to participate in races with purses worth less than $3,000 and drive alongside drivers of varying ages and talent levels.
Stewart was a spectator at the Knoxville Nationals in Iowa on Tuesday, the one-year anniversary of the accident, and posted on his Twitter account: "Thank you to everyone that worked so hard to get me back to where I'm at today. It's your life, live it!"
Roughly three hours after the accident in New York, Donny Schatz, a sprint car driver for Tony Stewart Racing, won the prestigious Knoxville Nationals in Iowa for an eighth time. Stewart had spent much of the earlier part of the week trackside in Iowa watching his drivers compete.
"I was just told there was an incident involving Tony. I don't know to what extent or what's happening," Schatz said.
The four-team Stewart-Haas Racing Sprint Cup organization fields cars for Stewart, Kevin Harvick, Kurt Busch and Danica Patrick. He's struggled a bit this year since returning, and heads into Sunday's race winless on the season and ranked 19th in the standings.
Stewart is scheduled to start 13th on Sunday at Watkins Glen International in south central New York state. He has just five races remaining to either score a win or move inside the top 16 in points to grab a valuable spot in NASCAR's championship race.
The Associated Press contributed to this report.