The tweet overstates the partisan makeup of the special counsel's team and falsely asserts that no Republicans are on it.
Mueller is a longtime Republican. He was nominated as FBI director in 2001 by a Republican president, George W. Bush, and was appointed special counsel by the Republican whom Trump picked to be deputy attorney general, Rod Rosenstein.
At least seven of the 17 lawyers brought on by Mueller - including James Quarles, Jeannie Rhee and Andrew Weissman - have donated to Democratic political candidates, five of them to Hillary Clinton, Trump's 2016 opponent.
Under instruction from his attorneys, Trump has in the past been careful not to publicly criticise Mueller by name or otherwise directly antagonise the special counsel, but rather to make more general criticisms. On Saturday night, in an apparent change of strategy, Trump for the first time tweeted the name of the special counsel.
"The Mueller probe should never have been started in that there was no collusion and there was no crime," Trump wrote. "It was based on fraudulent activities and a Fake Dossier paid for by Crooked Hillary and the DNC, and improperly used in FISA COURT for surveillance of my campaign. WITCH HUNT!"
The president's attack comes after his personal attorney, John Dowd, on Saturday called for an end to the Mueller investigation. He initially told the Daily Beast that he was speaking on behalf of the president, though he later backtracked and told The Washington Post that he was speaking only for himself.
Rep. Trey Gowdy, R-S.C., a member of the House Intelligence Committee, warned Trump that any interference in the Mueller probe would result in "a very, very long, bad 2018".
"If you have an innocent client, Mr Dowd, act like it," Gowdy said on "Fox News Sunday." "Russia attacked our country. Let special counsel Mueller figure that out."
Later, as if directly addressing the president, Gowdy said, "If you've done nothing wrong, you should want the investigation to be as fulsome and thorough as possible."
Former New Jersey Gov. Chris Christie, a Republican and an informal adviser to Trump, said it would be inappropriate for the president to try to fire Mueller.
The special counsel has "conducted this investigation so far with great integrity, without leaking and by showing results, and I don't think the president's going to fire somebody like that," Christie said on ABC's "This Week."
Sen. Lindsey Graham, R-S.C., warned that a move by Trump to order the firing of Mueller would be a step too far for lawmakers.
"If he tried to do that, that would be the beginning of the end of his presidency because we're a rule-of-law nation," Graham said on CNN's "State of the Union."
Sen. Richard Durbin of Illinois, the chamber's second-ranking Democrat, said he is concerned about "a constitutional crisis in this country".
"This president is engaged in desperate and reckless conduct to intimidate the law enforcement agencies in this country and to try and stop the special counsel," Durbin said on "Fox News Sunday."
"That is unacceptable in a democracy."
In a pair of separate tweets on Sunday, Trump also attacked Comey and McCabe, both regular foils to the president. Trump sent his Sunday tweets from the White House. He departed the residence shortly before 10 a.m., heading to the Trump National Golf Club in Northern Virginia. Aides would not say who the president's golfing partners might be.
McCabe, who was fired from the FBI late Friday night just hours before he was set to retire with full benefits, has kept contemporaneous notes of his interactions with Trump, according to two people familiar with his records. McCabe's memos could prove useful to Mueller's investigators in their obstruction-of-justice probe.
Trump tweeted, "Spent very little time with Andrew McCabe, but he never took notes when he was with me. I don't believe he made memos except to help his own agenda, probably at a later date. Same with lying James Comey. Can we call them Fake Memos?"
In an Oval Office meeting in May, Trump asked McCabe whom he had voted for in the 2016 election, several current and former US officials have told The Post, and he complained about the political donations McCabe's wife received for her failed 2015 Virginia state Senate campaign.
Comey also took contemporaneous notes of his interactions with Trump and confided in McCabe about those private conversations, including when Trump asked him for his loyalty.
Comey is publishing a memoir next month that is expected to detail his interactions with Trump and investigation of Russian interference, among other topics.
In a Sunday morning tweet, Trump accused Comey of lying in testimony to Congress when he was questioned by Senate Judiciary Committee Chairman Charles E. Grassley, R-Iowa.
The president wrote: "Wow, watch Comey lie under oath to Senator G when asked 'have you ever been an anonymous source ... or known someone else to be an anonymous source ...?' He said strongly 'never, no.' He lied as shown clearly on @foxandfriends."
Trump in the past has masqueraded as a fake publicist by the name of "John Miller" or "John Barron" to leak flattering or boastful details about himself to tabloid reporters.
Both Graham and Sen. Jeff Flake, R-Ariz., a frequent Trump critic, said on CNN that the Senate Judiciary Committee should hold a hearing on McCabe's firing and give Attorney General Jeff Sessions and McCabe opportunities to explain their actions.
The handling of McCabe's firing - he was ousted just hours before his 50th birthday on Sunday, at which point he would have been able to retire with full benefits - drew bipartisan disagreement on Sunday.
"I don't like the way it went down," Sen. Marco Rubio, R-Fla., said on NBC's "Meet the Press."
"I would have certainly done it differently."
Rubio added, "He should have been allowed to finish through the weekend."
