According to lawmakers and Mr. Rosenstein’s prepared remarks, the deputy attorney general offered little clarity about how related congressional inquiries may proceed in light of Mr. Rosenstein’s appointment on Wednesday of a special counsel, Robert S. Mueller III, to examine the possibility of collusion between President Trump’s associates and Russian officials.

The fact that Mr. Mueller’s inquiry is focused on possible crimes is almost certain to limit the cooperation of potential subjects of the investigation who might otherwise testify before Congress or share documents.

The decision by Mr. Comey to appear before the intelligence committee underscored the quandary of the competing claims between the imperatives of a special counsel and those of congressional committees.

Senator Lindsey Graham, Republican of South Carolina and a member of the Judiciary Committee, questioned Mr. Comey’s decision to testify, but then added that he believed he should also testify before his committee because it has oversight jurisdiction of the F.B.I.

The same sentiment seemed to be emerging in the House.

“Congress is going to want to look over the shoulder of this investigation,” said Representative Darrell Issa of California, a Republican who was an aggressive chairman of the House oversight committee during the Obama administration. “The executive branch will always try to limit that for fear it will contaminate potential criminal investigations, or leaks.”

“I don’t expect this to be any different,” he added.

Mr. Rosenstein instead delivered careful characterizations about the inquiry and deferred to Mr. Mueller’s autonomy as special counsel, and a pending investigation into Mr. Comey’s conduct by the Justice Department’s inspector general. He said that his memo had not been a legal brief, or a finding of official misconduct by Mr. Comey or “a statement of reasons to justify a for-cause termination.”

Representative Jackie Speier of California, a Democrat on the House Intelligence Committee, said Friday that Mr. Rosenstein had made it clear that Mr. Mueller would have “carte blanche authority,” but he also reassured lawmakers that he understood the role of Congress.