Rosenstein plans to brief members of the House at a closed-door meeting Friday.
Although he piled on Comey in the memo and called for new leadership, he stopped short of calling for his firing.
Mr Rosenstein said, although he was personally fond of Mr Comey, "I thought it was appropriate to seek a new leader". Trump fired Comey last week.
U.S. president Donald Trump has claimed he is the victim of an unprecedented "witch hunt" that "hurts our country terribly" in the wake of an independent probe into claims of Russian Federation links.
US President Donald Trump has denied all allegations that he tried to pressure fired FBI Director James Comey into dropping a Russia-related probe, and called the appointment of a special counsel to probe claims of his collusion with Russian Federation a "witch hunt".
The letter Rosenstein penned severely criticizing Comey's handling of the Hillary Clinton email investigation. "And the need for former Director Comey to come testify in public soon is as great as ever".
Then, days before the election, he told lawmakers the inquiry was gearing up again to look at Clinton emails that ended up on disgraced Rep. Anthony Weiner's computer.
During the question period, a reporter asked Trump if he "urged" Comey to "close or to back down the investigation into Michael Flynn?". (Rosenstein backed up this version, telling a Senate committee that Trump had already chose to fire Comey before he wrote his letter.) But after Trump saw the backlash to his seeming honesty, he and his staff backtracked to the recommendation line.
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, who served under George W. Bush and Barack Obama, preceding Comey as head of the Federal Bureau of Investigation.
The latest political storm, coupled with the still-potent fallout from Trump's recent disclosure of classified information to Russian diplomats at the White House, has overshadowed all else in the capital and beyond.
South Carolina Sen. Lindsay Graham took it a step further, saying that the Justice Department's look into the Trump campaign's possible Russian Federation ties is "now considered a criminal investigation". The career official had a reputation for integrity and bipartisan backing.
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".
The appointment of Mueller as special counsel has drawn generally favourable comments from Democrats and from some Republicans as well. "I don't think (Rosenstein) did a lot to bolster our confidence in him today".
Ultimately, he was confirmed by the Senate on a 95 to 6 vote on April 25.
US Speaker of the House Paul Ryan speaks to the press about President Donald Trump, former FBI Director James Comey and Russian Federation investigations as Republican Conference Chairman Rep. Cathy McMorris Rodgers (R) looks on after a closed meeting of the Republican leadership of the House of Representatives on Capitol Hill in Washington, US May 17, 2017. During that lunch, Comey described his efforts to keep the President and the White House at arm's length. In subsequent days, however, Trump acknowledged he'd made a decision to fire Comey even before he read Rosenstein's memo.
Durbin and other lawmakers said Rosenstein did not indicate who in the Trump administration told him to write the memo with the rationale for ousting Comey.
He says his memo is not a finding of official misconduct and is not a statement of reasons to justify Comey's firing.