Officials said Stone was in intensive care following the fighting early yesterday but expected to fully recover.
"We know it's not related to what occurred in France," Sacramento Deputy Police Chief Ken Bernard said.
Stone was out with four friends when they got into a fight near a bar with another group of people, Bernard said. The deputy chief would not disclose what sparked the argument but said there was no evidence the assailants knew who Stone was.
A grainy surveillance video from a camera outside a liquor store shows a large man who appears to be Stone fighting against a half-dozen people at an intersection as cars as onlookers pass by.
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The group spills into the street as people take swings at each other, and the man who appears to be Stone knocks one person down before another man strikes at his back. Police said two assailants fled in a car. No immediate arrests were made.
Kirk said Stone remained heavily sedated in the intensive care unit. He declined to discuss any details about the surgery or whether any vital organs were damaged in the stabbing, beyond saying Stone had "significant injuries."
The airman was conscious when he arrived at the hospital. "I suspect given his history of recent events he is quite a fighter," Kirk said.
Doctors expect Stone to fully recover. Stone's family asked Kirk to convey their gratitude for all the expressions of concern they had received.
He had boarded the train with a Kalashnikov rifle, pistol and box cutter.
Stone, who is assigned to Travis Air Force Base in California, suffered a severely cut thumb and a knife wound to his neck during the struggle with the gunman.