Mueller is the man Rosenstein tapped Wednesday to head the U.S. Justice Department's investigation into whether anyone on Trump's campaign colluded with Russian operatives who sought to interfere with the 2016 presidential election.
Trump deployed his characteristic social media bombast to take his unvarnished response directly to voters, elaborating on the staid written statement issued the night before that denied any collusion with Kremlin-linked officials.
"With all of the illegal acts that took place in the Clinton campaign & Obama Administration, there was never a special counsel appointed!" he said, without providing evidence for those claims. "This is the single greatest witch hunt of a politician in American history!" he Tweeted, also complaining that Hillary Clinton and President Obama were not investigated by a special counsel. "If Donald Trump broke it, he must be vigorously prosecuted without fear or favor", Jeffries said in a statement.
Rosenstein is the top official at the Justice Department responsible for overseeing the investigation after Jeff Sessions, Trump's attorney general, recused himself from the Russian Federation probe.
Comey was then fired by Trump as the investigation continued. The House and Senate intelligence committees are working on aspects of all that, and those must continue.But a full accounting is likely to emerge only if Congress appoints a special commission like the one that investigated the 9/11 attacks.
But Rosenstein's decision prompted further outrage from Trump.
The law on special counsel says the attorney general, or acting attorney general in cases where the attorney general is recused, can appoint a special counsel when a case presents a "conflict of interest" for the Justice Department, or "other extraordinary circumstances".
"The entire thing has been a witch hunt and there is no collusion between certainly myself and my campaign - but I can always speak for myself - and the Russians". Cory Gardner, R-Colo. "We need to get all the facts, and Mueller is the type of person that is respected and trusted by both Republicans and Democrats and will put country over politics". That would surely include Trump's firing last week of FBI Director James Comey.
Most Republicans did not join their Democratic colleagues in calling for a special prosecutor, preferring to keep their faith in the congressional and career FBI investigators.
The Plame investigation, for example, dogged the Bush administration from the end of its first term and through most of is second term until the president commuted Libby's prison sentence in a controversial decision.
The general feeling among Colorado lawmakers, however, was that Mueller's appointment was welcome and necessary.
Asked outright whether he had asked Comey - as reported - to drop the investigation into the Moscow connections of his sacked national security advisor, Michael Flynn, Trump fired back: "No".
Since then, the President had said nearly nothing in public about his legislative agenda, focused instead on firing the FBI Director, and dealing with a growing investigation into the Russian Federation matter.
U.S. Rep. Dan Donovan, a Republican representing South Brooklyn and Staten Island, said he had "complete confidence in the Federal Bureau of Investigation and Justice Department, along with the House and Senate committees".
The Senate intelligence committee also asked acting FBI Director Andrew McCabe to turn over any notes Comey might have made regarding discussions he had with White House or Justice Department officials about Russia's efforts to influence the election.