Trump strongly disagreed. The appointment, he said in a briefing with news anchors, "hurts our country terribly."
He said it "shows we're a divided, mixed-up, not unified country" and is "a very, very negative thing."
He leapt to make the point again at a joint news conference with President Juan Manuel Santos of Colombia, describing the development as a distraction.
"Well I respect the move, but the entire thing has been a witch hunt," he said, insisting there had been "no collusion" between his campaign and Russia.
"I'm fine with whatever people want to do," he added. "But we have to go back to running this country really, really well."
The Justice Department announced Wednesday that former FBI director Robert Mueller has been given sweeping power to investigate Russian interference in the 2016 presidential campaign, including potential links between Moscow and Trump campaign associates.
Despite initially opposing appointment of an independent counsel, House Speaker Paul Ryan said Thursday that the development "helps assure people and the Justice Department that they're going to go do their jobs independently and thoroughly, which is what we've called for all along."
But Trump, after issuing a measured statement when the news first broke Wednesday evening, allowed his resentment to burst forth Thursday in angry morning tweets.
"This is the single greatest witch hunt of a politician in American history!" Trump wrote, ignoring impeachment efforts and blistering verbal attacks on previous presidents and other political leaders.
"With all of the illegal acts that took place in the Clinton campaign & Obama Administration, there was never a special counsel appointed!" he added later, without providing examples.
Trump is leaving Friday for his first foreign trip, to the Mideast and beyond, which aides hope can have the effect of refocusing a White House in disarray.
The president's tweets and comments to the TV anchors drew little reaction from fellow Republicans, who instead joined Democrats in heaping praise on Mueller, a longtime respected lawman who served under George W. Bush and Barack Obama, preceding Comey as head of the FBI. Now Mueller will have nearly unfettered access to witnesses and information, and the ability to bring criminal charges.
His appointment raises the stakes dramatically on the long-simmering allegations that Russia meddled in the 2016 election and had connections with members of the Trump campaign.
Democratic senators had been prepared to press Rosenstein Thursday to take the step of appointing a special prosecutor, but were left praising him instead before his closed-door briefing began.
"This was a very good first step. Mr. Rosenstein has done the right thing," Senate Minority Leader Chuck Schumer of New York said on the Senate floor. "I now have significantly greater confidence that the investigation will follow the facts wherever they lead."
Despite the appointment, at least three congressional committees are continuing their investigations, leading to some turf warfare and sniping as the Senate Intelligence Committee and the Senate Judiciary Committee both sought to lay claim to testimony from Comey, while the House Oversight Committee also hoped to hear from the former director.
On a day of fast-moving developments, the House intelligence committee announced that it, too, had asked for documents, in this case from the FBI and the Justice Department.
The No. 2 Senate Republican, John Cornyn of Texas, said he is supportive of investigations in Congress but expressed concern about the "proliferation" of hearings. "I hope that we don't inadvertently trip up or damage the independent investigation of the special counsel," he said.
There was confusion during the day surrounding Flynn, Trump's former national security adviser who has emerged as a central figure because of his own ties to Russia, which led to his dismissal early on in the Trump administration. Intelligence Committee Chairman Richard Burr indicated at one point that Flynn was resisting the committee's document subpoenas, but later clarified that he hadn't actually heard from Flynn's lawyer to that effect and he would welcome "their willingness to cooperate." It was not clear what caused the mix-up.