In an interview with Empire, the actor and former California governor claimed that low ratings were due to President Trump's credit as executive producer.
He said he would like to work "on a show that doesn't have this baggage".
However, Mr Trump tweeted that Schwarzenegger "isn't voluntarily leaving".
He said the actor was "fired by his bad (pathetic) ratings", adding that it was a "sad end to [a] great show" - referencing his own previous tenure as its presenter.
As the row escalated, Schwarzenegger replied: "You should think about hiring a new joke writer and a fact checker."
The president - who left the show to run for the White House - has previously ridiculed his TV boardroom successor.
In Friday's interview, Mr Schwarzenegger claimed an anti-Trump social media campaign had impacted the ratings of the show, on which he debuted in January.
"When people found out that Trump was still involved as executive producer and was still receiving money from the show, then half the people [started] boycotting it," said the former Mr Universe.
"It's not about the show... because everyone I ran into came up to me and said, 'I love the show... but I turned it off because as soon as I read Trump's name I'm outta there!'
"It's a very divisive period now and I think this show got caught up in all that division," he said, adding that he would decline if asked to host again."
Mr Schwarzenegger has often clashed on Twitter with Mr Trump.
Last month at the National Prayer Breakfast in Washington DC, Mr Trump mockingly asked people to "pray for Arnold... for those ratings".
He said the show had been swamped "in comparison to the ratings machine, DJT", using his own initials.
The show, whose 15th season ended last month, averaged fewer than five million viewers per episode.
New York real estate businessman Mr Trump became a household name as the show's host for 14 seasons.
Some 20 million Americans were regular watchers in its first year before the viewing figures tapered to six million for Mr Trump's last episode.
Arnold Schwarzenegger has quit The New Celebrity Apprentice, saying it has become tainted by Donald Trump's involvement.