Trump said the rising talk of impeachment on Capitol Hill "totally ridiculous", and flatly denied that he asked former FBI Director James Comey to end the agency's probe into former National Security Adviser Michael Flynn.
Trump was meeting with FBI director candidates when the White House was formally told that a special prosecutor had been named in the Russian Federation investigation. Trump fired Comey on May 9.
Trump spoke at a meeting with top news anchors at the White House.
Rosenstein briefed the US Senate just one day after he turned the political world on its head by appointing a special counsel to investigate possible collusion between the campaign of President Donald Trump and Russian officials meddling in last year's election.
The news conference marked the first substantive public comments by the President in the last two weeks, since he celebrated the May 4 approval in the House of a GOP health care bill.
Trump said Thursday that he respects "the entire thing" but said that he personally has no ties to Russian Federation.
After the White House put out a subdued statement on Wednesday night about former FBI Director Robert Mueller being named special counsel, Trump publicly vented on Thursday morning about the new probe.
"With all of the illegal acts that took place in the Clinton campaign & Obama Administration, there was never a special (counsel) appointed", Trump tweeted. Trump later deleted and retweeted that statement to correct the spelling of "counsel". "This is the single greatest witch hunt of a politician in American history".
"This is a truth hunt", said Democratic Senator Amy Klobuchar.
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 favouring economic growth.
Mueller, he explained, will be "completely siloed off from the Department of Justice and the Federal Bureau of Investigation".More news: Humans Accidentally Created Protective Barrier Around Earth
More news: President Trump to meet 4 candidates for post of Federal Bureau of Investigation director
More news: With 'tapes' tweet, Trump evokes Nixon's White House
Rosenstein said Mueller's appointment as special counsel - also known as a special prosecutor - does not mean "a finding that crimes have been committed or that any prosecution is warranted".
Afterward, Republican Senator Lindsey Graham told reporters that "everything he said was that you need to treat this investigation as if it may be a criminal investigation".
A self-described friend of Comey's wrote in a public blog post on Thursday that Comey had told him that he had rebuffed a Trump request for loyalty by promising only honesty.
By Wednesday evening, a Republican source said that the move effectively shut down Congress' intense digging into Comey and his conversations with Trump.
Sen. Dick Durbin, D-Ill., also said Rosenstein learned on May 8 that Trump "was going to terminate Comey".
This came days before Trump is about to embark on his first foreign trip to the Middle East and Europe, and amid questions of whether the USA leader has lost his credibility and leverage among foreign partners.
But with Trump, said a Republican who formerly worked in the White House, "the trouble will be if he goes off script. Trump can't control himself".
Though the Mueller appointment has obviously raised the legal and political stakes for Trump, it is widely thought that Trump is unlikely to be impeached over the current political turmoil.
It later emerged that Comey had made notes of his meetings with Trump, who reportedly asked him to quash an investigation into Flynn's contacts with foreign governments.
The surprise announcement to hand the probe over to Mueller, a lawman with deep bipartisan respect, was a striking shift for Trump's Justice Department, which had resisted increasingly loud calls from Democrats for an outside prosecutor.
House Speaker Paul Ryan said the appointment was consistent with his goal of ensuring that "thorough and independent investigations are allowed to follow the facts wherever they may lead".