Peter Nunn, was sentenced to 18 weeks in jail last year for sending indecent, obscene or menacing messages.
Nunn, who has already served his sentence, had lodged an appeal which was due to be heard at the Old Bailey.
However, the 34-year-old, part-time delivery driver from Bristol backed out at the last minute.
The court heard his circumstances had changed "quite considerably" since his release.
Nunn's lawyer, David Patience, said he decided not to go ahead because his mother had been sick and he had childcare commitments after his partner had broken up with him following the trial.
He said his client wanted to abandon both the appeal against conviction and appeal against sentence.
The judge said he would not order him to pay costs but warned Nunn he must continue to abide by a restraining order to stay away from Ms Creasy or face prison again.
A man who sent abusive Twitter messages to politician Stella Creasy has abandoned an appeal against his conviction.