In a statement, the presenter said he would continue to present his programmes on BBC Radio 2 and Greatest Hits Radio.
The Radio 2 presenter says he would "not wish on anyone" the experience of interrupting breeding foxes.
Steve Wright will host Pick of the Pops, while its presenter Paul Gambaccinni moves to Sundays.
Broadcaster Paul Gambaccini strongly criticises Cressida Dick after his treatment by the Metropolitan Police.
