Burger suffered the injury during Saturday's 29-15 victory over Scotland in Port Elizabeth and later complained of numbness down one side of his body.
Burger has received three different opinions with regard to his recovery time, ranging from three weeks to six months.
"This morning I was told Schalk would be out for three weeks," White said on Monday.
"Then he saw someone else and I was told it would be eight weeks if he had an operation. Now I've just been told that he will be out until January.
"I'm scared to let him go to another doctor, because he might say that Schalk can never play rugby again."
Burger apparently injured himself in the 18th minute when he collided with a Scottish player's hip. He continued to play, however, and only realised after the game that something was amiss.
"Schalk got injured early on, but didn't think it was too serious," added White.
"On the plane trip to Cape Town, he said he was sore - and if he says he's sore then he's really sore. He has the highest pain threshold of any player in the squad.
"He complained that his entire body was sore and when he started getting pins and needles down one side, we got very worried."
According to White, Burger may require a neck fusion operation, which may threaten his career. White has decided, however, not to call up a replacement despite speculation that Cats captain Wikus van Heerden would get the nod.
"They have to take bone from his hip," said White. "I don't want to jump to conclusions, but I suppose there's a chance he may never play again.
"This again backs up what I've been saying for ages. We have to manage our players more carefully. Schalk is 23 years old and needs a major operation.
"He's going to get six months' rest during the Test season when it should be the other way around.
"Look how many of the 2002 South Africa Under-21 side aren't playing now - Ashwin (Willemse), Gurthro (Steenkamp), Gordon Gilfilan had to have a cruciate ligament replacement, Clyde Rathbone was out for a year, Ricky Januarie missed last year's end of year tour, Juan Smith has been hurt - the list just goes on and on.
"We need to develop our players properly physically.
"Yes, Schalk's was a freak injury, but so was Ashwin's and so was Gurthro's. The fact is 22 and 23-year-olds are picking up freak injuries playing against 28-year-olds."
Burger is not the only injury woe for the Springboks ahead of Saturday's one-off Test against France. Midfielder Andre Snyman is suffering from a wrist injury and winger Breyton Paulse has a hamstring strain.
Lock Bakkies Botha was back in training on Monday while injured midfielder Jaque Fourie also rejoined his team-mates despite a cracked rib, which is said to be healing fast.
The Boks have also lost first-choice inside-centre Jean de Villiers to a bruised rib.