The Jewish Labour Movement also called the decision "shocking and saddening", saying he had been a "strong ally in the fight against anti-Semitism in the Labour Party".
Tom Watson steps down as Labour Deputy Leader
Labour's deputy leader Tom Watson is stepping down from his role amid turmoil in Labour's ranks over the continuing antisemitism with the party.
Mr Watson has often been at odds with the leadership and faced an attempt to oust him at Labour's conference.
As an ardent Remainer, Mr Watson was also at odds with his own constituency, which voted 66% in favour of Leave at the 2016 referendum.
In his letter to Mr Corbyn, the former MP for West Bromwich East thanked the leader "for the decency and courtesy you have shown me over the last four years, even in difficult times".
He added: "Our many shared interests are less well known than our political differences, but I will continue to devote myself to the things we often talk about" - including gambling regulation, stopping press intrusion and campaigns on public health.
He told the BBC that after 35 years in front-line politics, he wanted to "take a leap and do something new", but he said he would be out campaigning for the Labour Party to make sure a Labour government is elected.
In recent months, Mr Watson has also faced criticism for meeting Carl Beech, the paedophile fantasist who falsely accused VIPs of sexually abusing him.