The partnership between the two men began when Musk endorsed Trump last July after an assassination attempt in Pennsylvania. The Tesla boss reportedly funnelled $290m (£213m) into getting him back into the White House.
Amid a flurry of posts on X after Thursday's news conference, Musk took credit for the sweeping Republican victory in last November's election, writing: "Without me, Trump would have lost the election, Dems would control the House and the Republicans would be 51-49 in the Senate."
"Such ingratitude," he added.
Musk went on to post a poll, asking his followers: "Is it time to create a new political party in America that actually represents the 80% in the middle?"
Over the course of the day, Musk went on to repost a tweet calling for Trump to resign, argue that his global tariff plan will trigger a US recession, and to suggest without evidence that Trump appears in unreleased files related to late sex offender Jeffrey Epstein.
Epstein was arrested in July 2019 on charges of sex trafficking and died by suicide while awaiting trial. Trump was president at the time. He said he knew Epstein "like everybody in Palm Beach knew him" but had a "falling out with him a long time ago".
The White House condemned Musk's allegation, with press secretary Karoline Leavitt saying in a statement: "This is an unfortunate episode from Elon, who is unhappy with the One Big Beautiful Bill because it does not include the policies he wanted."
On his Truth Social network, Trump claimed that Musk "just went CRAZY" and went on to post: "The easiest way to save money in our Budget, Billions and Billions of Dollars, is to terminate Elon's Governmental Subsidies and Contracts. I was always surprised that Biden didn't do it!"
Musk's companies, including Tesla, SpaceX and Starlink have direct contacts with the US government and, like many other businesses, also benefit from subsidies and tax breaks.
In response, Musk said SpaceX "will begin decommissioning its Dragon spacecraft immediately". The craft is used to shuttle people and supplies to the International Space Station.
However, he later he appeared to back down from that threat, saying in response to a post on X urging him to cool off: "Ok, we won't decommission Dragon."
Telsa stock dropped by 14% within hours of the row bursting out into public.
According to the most recent analysis from the nonpartisan Congressional Budget Office, the spending bill working its way through Congress will increase the US national debt by $2.4tn over 10 years and leave nearly 11 million people without government-backed health insurance.
The White House disputes those figures, saying they don't account for revenues brought in by increased tariffs.
Put in charge of radically slashing government spending at Doge, Musk initiated mass sackings and wholesale elimination of departments such as the US Agency for International Development (USAID).
Doge claims to have saved $180bn, although that number has been disputed, and is well short of Musk's initial aim to cut spending by up to $2tn.