At a press conference later on Thursday, Trump said, "I respect the move but the entire thing is a witch hunt".
Trump himself, in a statement right after the appointment late Wednesday, said merely, "A thorough investigation will confirm what we already know - there was no collusion between my campaign and any foreign entity".
"I accept this responsibility and will discharge it to the best of my ability", he said in a statement.
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"This is the single greatest witch hunt of a politician in American history", he tweeted on Thursday morning.
The Russia issue has, however, clouded his early months in office.
USA intelligence agencies believe Moscow tried to tip the election in favour of Mr Trump.
"The appointment of former FBI director and respected lawyer Robert Mueller as special counsel for the Russian Federation investigation is a positive development and will provide some certainty for the American people that the investigation will proceed fairly and free of political influence", Senator Richard Burr said and Mark Warner, chairman and ranking member of Senate Select Committee on Intelligence, said.
USA stocks recovered ground on Thursday as upbeat economic data emboldened investors to return to the market, a day after Wall Street saw the biggest selloff in eight months on worries the political turmoil could undermine Trump initiatives such as tax cuts that investors see as favoring economic growth.
On a day of fast-moving developments, the House intelligence committee announced that it, too, had asked for documents, in this case from the Federal Bureau of Investigation and the Justice Department. Tim Scott, R-S.C. "I do think that the special prosecutor provides a sense of calm and confidence perhaps for the American people, which is incredibly important".
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Macron will also become the first President from outside the two traditional main parties since the modern republic's foundation in 1958.
To emphasize the independence of his decision, Rosenstein did not notify White House counsel Donald McGahn of the appointment until 5:30 p.m., the same time Justice Department officials were briefing reporters and 30 minutes before the news became public. He was sacked as national security adviser in February after it emerged he had misled Vice President Mike Pence about conversations with the Russian ambassador to Washington in December.
Lawmakers on Capitol Hill generally welcomed the Justice Department action and praised Mueller for his integrity, but House and Senate Republican leaders said they would go on with their own investigations of the Russian Federation matter. "With this appointment, my hope is that the distractions can stop, the counsel can do his job and Congress and the President can deal with the many challenges facing our country".
The appointment was widely endorsed by politicians from both sides.
Mueller, 72, was decorated as a Marine Corps officer during the Vietnam War.
Mueller, a former Marine officer and federal prosecutor, was sworn in as the sixth director of the FBI on September 4, 2001 - just a week before the September 11 terrorist attacks.
He served beyond the normal 10-year term for directors in order to allow time for Comey to be confirmed.
The chairman of the Senate Intelligence Committee, Republican Richard Burr, said Flynn's lawyer told the panel he would not honour its subpoena seeking relevant documents. Whatever the reason, Rosenstein would have to inform Mueller in writing. "We still need an independent investigation", the Democratic National Committee blasted to its email list. Probably the most famous (or infamous) such figure is Kenneth Starr, who was initially appointed to lead an inquiry into a real estate deal involving the Clintons, and ended up exposing former US president Bill Clinton's affair with Monica Lewinsky.
Meanwhile, Senate Democrats met in private Wednesday to brainstorm how they were going to grill Rosenstein - including pressing him to name a special prosecutor.
The FBI, with the help of the Treasury Department, the CIA and other agencies, is examining evidence of possible contacts, money transfers and business relationships between a variety of Trump associates and Russian officials, the sources say...
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