Pickering, who helped both GBR1 and GBR2 sleds qualify for the Games after being recruited from track and field, sustained an acute disc prolapse, while training at an Olympic preparation camp in Konigsee, Germany.
The 27-year-old, who won a bronze medal in the 4x100m relay at the 2007 world athletics championships, also missed the 2012 London Olympics because of a back injury.
"I am devastated to have suffered a similar fate just days before the start of Sochi 2014 Games," he said in a statement.
"I will cheering the boys on from the UK and wish the rest of Team GB the best of luck in Sochi."
(Reporting by Justin Palmer, editing by Mitch Phillips)
