The soldier, 28, was also left with injuries to his face, a fractured cheekbone and eye sockets and a broken nose, the Metropolitan Police said in an appeal launched on Thursday.
He required extensive surgery following the attack close to Piccadilly Circus, in central London.
The force released a CCTV still of a man they are trying to trace after the attack at 3.20am on Thursday January 31 on Haymarket, at the junction with Coventry Street.
In the picture he is bearded with short, dark hair and appears to be wearing a black sweater and blue jeans or trousers.
Anyone who recognises the man should call Westminster CID on 101, quoting Cad 675/31Jan, Tweet @MetCC or call Crimestoppers anonymously on 0800 555111.