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.

The female swan had a netted bag, used to hold footballs, wrapped around her head and wings at Wyken Slough nature reserve in Coventry.
She remains at Wychbold Swan Rescue, near Droitwich, Worcestershire, following Saturday's rescue.
The RSPCA said fishing line and hooks were also "dangerous hazards".
The swan, which had been seen struggling to keep her head above water, sustained damage to one of her wings and remained under observation on Friday.
RSPCA inspector Helen Smith said the swan was "in complete shock" when she was rescued by members of the West Midlands Fire Service.
"She could have been struggling to free herself for some time and was clearly exhausted," she said.
"The netting... was so tightly wrapped around her body she was in real danger of drowning."
The RSPCA inspector reminded members of the public about dangers to wildlife and wild birds.
She said: "Netting, litter and fishing line and hooks are dangerous hazards to wildlife and I urge people to think of the consequences before they casually discard such items."

Fridays Ltd, which produces thousands of free range eggs in Cranbrook, wants to expand its operation in Horsmonden.
Residents fear the extra traffic and noise from the 32-hectare site will have a detrimental impact on the area.
The company said it was responding to a 5% year-on-year increase in demand for free range eggs.
The family-run business is one of the three biggest egg producers and packers in the country.
If agreed, its new chicken farm would be located at Pullens Farm, in Lamberhurst Road, which is on the outskirts of the village of Horsmonden.
Resident Sarah Overton said: "The sheer scale of this development is extraordinary.
"We're talking the equivalent of 80 football pitches, 64,000 birds at any one time producing 360,000 eggs a week, using mechanical processes and only needing one full-time person to look after them, and three part-time egg collectors.
"It equates to industrialisation in the countryside, which is bad enough, but this is industrialisation of an area of outstanding natural beauty with protected species that we all need to be careful to preserve," she said.
Fridays Ltd has lodged its application with Tunbridge Wells Borough Council, with a provisional decision expected in August.
Horsmonden Parish Council is discussing the plans on Monday.
Arnold Schwarzenegger has quit The New Celebrity Apprentice, saying it has become tainted by Donald Trump's involvement.
[NEXT_CONCEPT]
A warning has been issued about the dangers of discarded netting and litter after the rescue of a swan which was in danger of drowning.
[NEXT_CONCEPT]
Plans by a company that wants to put 64,000 chickens on an area of outstanding natural beauty in Kent have been met with anger by local residents.