Dr Billings called for Mr Crompton's resignation when he seemingly justified questioning the behaviour of fans after the Hillsborough disaster.
Mr Compton was commenting after a jury concluded that police actions contributed to or were the cause of the disaster which killed 96 fans.
The High Court ruled that Mr Crompton should not have been forced to resign by Dr Billings.
Dr Billings said: “I will not be appealing the decision. Instead, I will be supporting the Association of Police and Crime Commissioners in dealing with the constitutional issues raised by the judgement through Parliamentary means rather than engaging in an expensive legal process.
"I don't believe it is in the interests of the South Yorkshire public to commit any further funds and I feel that what South Yorkshire Police needs now is a period of stability.”