The White House, however, said initially that it was Rosenstein's memo on May 9 that led to Mr. Trump's decision to fire Comey last week.
Trump was speaking in a joint press conference with Colombia's President Juan Manuel Santos.
Trump said later, however, that he had been thinking about "this Russian Federation thing" when he fired Comey, whom he called a "showboat" and a "grandstander" in an interview with NBC's Lester Holt last Thursday. "There is no collusion between certainly myself and my campaign".
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. "So I can tell you that we want to bring this great country of ours together".
Trump called talk of potential criminal charges or impeachment "totally ridiculous".
He also said he thought that Comey's firing would be bipartisan.
"I think it is totally ridiculous".
Republican Senator Lindsey Graham of SC told reporters that it appears the FBI investigation is turning into a criminal investigation rather than a counterintelligence probe.
Trump won't move US Embassy to Jerusalem for now, senior official says
During the time between surveys, the president has welcomed Palestinian Authority leader Mahmoud Abbas to the White House. After negotiating that diplomatic minefield, Trump is scheduled to visit Yad Vashem, Israel's Holocaust Memorial.
"That was very reassuring", he said, adding that the scope of the probe will be "so broad" that Rosenstein refrained from commenting on too much.
"I believe that's why the deputy attorney general went out and wrote his very very strong letter", he said. "When I made that decision I actually thought it would be a bipartisan decision because you look at all of the people on the Democratic side, not only the Republican side, that were saying such awful thing about Director Comey".
President Donald Trump flatly denied that he asked former FBI Director James Comey to end his investigation into Trump's former national security adviser Michael Flynn.
"I hope you can let this go", Comey wrote, quoting Trump in the document, which CNN has not viewed but which was described by the sources.
The New York Times had recently reported on a memo, allegedly written by Mr Comey, claiming Mr Trump had asked him to drop the investigation during a private meeting at the White House.
Reacting to the president's Thursday morning tweets, Rep. Walter Jones (R-N.C.) joked, "I wish the president would stop tweeting, and that he would listen to the birds, maybe, but not create the sound himself".
Sen. Lindsey Graham, R-South Carolina, said, "It's now considered a criminal investigation and Congress's ability to conduct investigations of all things Russian Federation has been severely limited - probably in an appropriate way".
"No politician in history, and I say this with great surety, has been treated worse or more unfairly", he said on Wednesday to the graduating class of the United States Coast Guard Academy. But Graham emphasized "the takeaway I have is that everything he said was that you need to treat this investigation as though it may be a criminal investigation".