Walker's parents traveled to Oklahoma to be with their son on Saturday night. His brother told the Associated Press that his mother watched the game on television when the accident occurred.
And while his teammates returned to New Orleans, members of the Tulane athletic staff remained in Oklahoma.
Stewart, who accompanied Walker in the ambulance to the hospital, said that Walker was alert and breathing on his own.
"His neck hurt but he was alert, and again, a lot of what was going on during that time was us just telling him to relax," he said. "As you can imagine he's scared with all this."
The accident occurred in the final play of the first half of the Conference USA opening game, when Tulsa led 35-3. The final score was 45-10.
It may be weeks or even years before doctors fully understand the severity of Walker's injury and how likely his recovery might be, said Dr. Arthur Jenkins, who specializes in studying and treating spinal injuries at Mt. Sinai Hospital in New York City.
"It also could be years before his final degree of neurologic recovery becomes clear," Jenkins told ABC News. "Even severely neurologically impaired patients can improve function over a long time period."
"The severity of the injury is determined by the amount of tissue damage that is permanent," he added.
Jenkins, who has researched and treated spinal injuries for more than 15 years, said that at this point, Walker has most likely reached the peak period for spinal swelling, which will give his doctors critical signs about how well he may recover.
"Typically you want to see how they wake up from the operation. The second thing you'll do is watch how he does in the first two days after the operation," Jenkins said. "A neurologic injury tends to be at its worse about three days after the injury occurs."
The mood was a "somber" one for Walker's teammates as they traveled back to New Orleans after the game, Stewart said.
"There's an understanding that this could turn out bad, we don't know," Stewart said. "And anytime that anybody is taken off a field in an ambulance—whether it's for a broken foot nor a neck injury—it's very concerning to everybody."
"These are young kids who think they are invincible. It brings home a bit of real world reality to it," he said.