Trump blasted that decision, echoing his Thursday morning tweets and comments he made over lunch with network news anchors that the Russian Federation investigation is a "witch hunt" that "divides the country".
The appointment puts Mueller at the head of the probe into Russia's alleged meddling in the 2016 presidential election.
Former US officials told Reuters that Michael Flynn and other advisors had ties with Moscow.
Flynn was forced to resign on February 13 amid questions over his contact with the Russian ambassador to the United States and discussions of US sanctions.
"No. No", Trump told reporters during a news conference, "Next question".
Mr. Trump is scheduled to depart for his first foreign trip as president Friday.
The most likely option (45 per cent) is a Trump exit in 2020 or later, after the end of his first term.
"At best, President Trump has committed a grave abuse of executive power", House Minority Leader Nancy Pelosi of California said in a statement.
A spokesman for Ryan said that the exchange never happened. What's needed is for patriotic-minded Republicans to impose on him a professional chief of staff - a Rob Portman, or Mitch Daniels - and to get rid of the amateur hour hangers-on, beginning with Trump's own family members.
Pittsburgh Steelers cut tight end Ladarius Green
He ended up catching 18 passes for 308 yards and a touchdown before leaving a December victory over Cincinnati with a concussion. Green (concussion) was released by the Steelers on Thursday following a failed physical, Teresa Varley of Steelers.com reports.
Mr McCarthy added on Twitter: "This was an attempt at humor gone wrong. People in Washington will only take it as some new provocation or trick from the Kremlin".
Trump also repeatedly bashed his former Federal Bureau of Investigation director, saying he "was very unpopular with most people" and claiming that Rosenstein had made a decision to write a memo strongly criticizing Comey that Trump used as justification to fire him after Comey had a poor showing at a Senate hearing.
"My decision is not a finding that crimes have been committed or that any prosecution is warranted".
"There's been no collusion between certainly myself and my campaign, but I can always speak for myself, and the Russians. It does raise a lot of questions and I think, like most American and most Wisconsinites, I just want the truth". That means a thorough - and dispassionate - public investigation.
A few Republicans have began to call for an independent probe.
"It's a position of conscience for me", he said.
I have not seen any evidence of actual collusion. And this is where I stand.
On Monday, The Post reported that Trump discussed classified information with Russian officials.
The top cop, who was abruptly fired by Trump last week, had reportedly grown concerned over a number of interactions with Trump and other White House officials that he felt compromised the FBI's autonomy.