"Obviously we're relieved that James is awake and out of surgery," Schmidt Peterson Motorsports team owner Sam Schmidt said.
"That's the most important thing on our minds right now and we will do absolutely everything required to ensure a complete recovery."
It was the fourth frightening wreck during practice for the Indianapolis 500 in the past six days.
A part seemed to break about 50 minutes into practice and the driver from Oakville, Ont., spun hard into the wall. As his car slid back down toward the apron, it briefly slid on its right side and almost flipped before coming came to rest upright.
Hinchcliffe's left thigh was pierced by the car's right front rocker and the piece of equipment needed to be removed, according to two people familiar with the crash who spoke on condition of anonymity because IndyCar and Schmidt Peterson Motorsports were not releasing details of the injury.
The surgery at Indiana University Methodist Hospital was also needed to stop massive bleeding.
"It's gut-wrenching," said defending Indy 500 winner Ryan Hunter-Reay, a teammate of Hinchcliffe's until this season. "It's terrible. There's no other way you can think about it."
IndyCar announced it would resume practice nearly three hours after the crash while it investigated yet another incident that has sharply raised concerns about safety with the famous race scheduled for Sunday.
Helio Castroneves and Josef Newgarden went airborne in crashes last week that raised questions about the impact of new oval aero kit packages on the cars.
And on Sunday, Ed Carpenter flipped his car on its side in practice. IndyCar officials huddled with teams and the two manufacturers before announcing changes to reduce power and speeds before Sunday's qualifying session.
Scott Dixon of New Zealand took his second Indy pole with a four-lap average of 226.760 mph Sunday — and there were no wrecks.
All the cars are using the new aero kits, but Castroneves, Newgarden and Carpenter were all driving Chevrolets. Hinchcliffe was in a Honda, and it's the first Honda to have its wheels leave the track's surface during practice.
Hinchcliffe's right front suspension failed, according to Honda spokesman Dan Layton. Layton said Honda officials aren't concerned about the safety of its aero kit, but are more concerned about what caused the suspension problem as the Indianapolis 500 nears.
"It was shocking," driver Townsend Bell said. "The worst possible accident that you can have here is to be fully loaded right in the middle of the corner and have something break. It was a really horrible circumstance for him."
Derrick Walker, IndyCar's president of competition and operations, has blamed Castroneves' wreck on an aero balance setting that was pushed too far, Newgarden's on a cut tire and said Carpenter simply had an accident.
A year ago, Hinchcliffe sustained a concussion when struck in the head by debris from another car in the inaugural Grand Prix of Indianapolis on May 10. After the road course race, he missed five days while waiting to be cleared and then qualified second, despite limited practice time.
Hinchcliffe, who won at New Orleans last month and is eighth in the driver standings, qualified 24th for this year's 500.
Auto Racing Writer Jenna Fryer contributed to this report.