Ryan said the tussle over document requests between congressional Republicans and Rosenstein, who oversees the federal Trump-Russia investigation, doesn't rise to the level of "high crimes and misdemeanors" that could warrant impeachment under the Constitution.
"I don't think we should be cavalier with this process or with this term," Ryan said. He also said he is encouraged by progress on the document production.
Ryan made the comments a day after the group of 11 House Republicans sharply escalated the extended clash with the Justice Department by filing articles of impeachment against Rosenstein, who oversees special counsel Robert Mueller's investigation.
Their move late Wednesday came after months of criticism aimed at the department — and the Russia investigation in particular — from President Donald Trump and his Republican allies in Congress. Trump has fumed about Mueller's probe and has repeatedly called it a "witch hunt," a refrain echoed by some of the lawmakers.
Mueller is investigating Russian interference in the 2016 election and whether Trump's campaign was involved.
The impeachment effort is led by North Carolina Rep. Mark Meadows, the chairman of the conservative House Freedom Caucus who talks to Trump frequently and often defends him to colleagues.
After Ryan voiced his disapproval, Meadows acknowledged that he didn't currently have the votes to pass the impeachment resolution. The House left Thursday afternoon for a five-week recess.
Instead, Meadows said he had discussed a plan with Judiciary Committee Chairman Robert Goodlatte and the No. 3 House GOP leader, Rep. Steve Scalise of Louisiana, to vote Rosenstein in contempt of Congress if the department has not produced certain documents by the time the House returns in September.
Meadows said that would give the department "one last chance" before lawmakers moved to hold Rosenstein in contempt or impeach him.
Ryan's tone was far different.
"We do not have full compliance, and we have to get full compliance, but we have been making tremendous progress to that point," he said.
Meanwhile, Attorney General Jeff Sessions defended Rosenstein in a speech in Boston, saying he has the "highest confidence" in his top deputy. Rosenstein has overseen the Russia investigation since last year, when Sessions recused himself from the probe following reports of his own meeting with the Russian ambassador.
Asked in May about rumblings that House Republicans might move to impeachment, Rosenstein was defiant.
"I think they should understand by now, the Department of Justice is not going to be extorted," he said.
Meadows, Ohio Rep. Jim Jordan and the other Republicans who introduced the resolution have criticized Rosenstein and Justice Department officials as not being responsive enough as House committees have requested documents related to the beginning of the Russia investigation and a now-closed investigation into Democrat Hillary Clinton's emails.
The five articles would charge Rosenstein with failing to produce information to the committees, though the department has provided lawmakers with more than 800,000 documents. The resolution notes that Rosenstein was one of several department officials who approved what some Republicans say was improper surveillance of Trump campaign adviser Carter Page.
The impeachment resolution also criticizes Rosenstein for refusing to produce a memo that outlines the scope of the investigation and questions whether the investigation was started on legitimate grounds.
It is highly unusual, if not unprecedented, for lawmakers to demand documents that are part of an ongoing criminal investigation.
At a House hearing last month, Rosenstein, along with FBI Director Christopher Wray, faced angry Republicans who alleged bias at the FBI and suggested the department has conspired against Trump. Still, some GOP lawmakers said they would draw the line at impeachment.
"Impeachment is a punishment; it's not a remedy," said House Oversight and Government Reform Chairman Trey Gowdy, who has led one of several House GOP investigations into the department and has complained of anti-Trump bias. "If you are looking for documents, then you want compliance, and you want whatever moves you toward compliance."
The impeachment resolution came about two hours after Meadows, Gowdy, Goodlatte and other GOP lawmakers met with Justice Department officials about the documents. The department has created new search systems, set up classified reading rooms and tasked dozens of employees to produce the hundreds of thousands of documents that Republican lawmakers have requested over the past year.
Democrats on the House Judiciary, Oversight and Government Reform and intelligence committees called the impeachment effort a "panicked and dangerous attempt to undermine an ongoing criminal investigation in an effort to protect President Trump as the walls are closing in around him and his associates."
So far, the special counsel has charged 32 people and three companies. That includes four Trump campaign advisers and 12 Russian intelligence officers.
Democratic Reps. Jerrold Nadler of New York, Elijah Cummings of Maryland and Adam Schiff of California said Rosenstein "stands as one of the few restraints against the overreaches of the president and his allies in Congress."
In addition to Meadows and Jordan, the Republican lawmakers who sponsored the impeachment articles are Reps. Jody Hice of Georgia, Jeff Duncan of South Carolina, Andy Biggs of Arizona, Louie Gohmert of Texas, Matt Gaetz of Florida, Scott DesJarlais of Tennessee, Paul Gosar of Arizona, Bill Posey of Florida and Scott Perry of Pennsylvania.
Associated Press writers Alan Fram, Kevin Freking and Lisa Mascaro in Washington and Philip Marcelo in Boston contributed to this report.