Beardsley has been on leave since an investigation into racism and bullying allegations was launched last year.
The 58-year-old, who had been under-23s coach at Newcastle since 2009, had "categorically denied the allegations".
The former England international, who made 326 appearances for Newcastle as a player, said the "time was right to seek a new challenge".
Newcastle added it was "grateful for the contribution Peter has given over the years as a player, coach and ambassador and we wish him well for the future".
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