The party was unable to win enough support from its own senators for a bill to repeal the 2010 Affordable Care Act and decided not to put it to a vote.
US Senate finance committee chairman Orrin Hatch said on Monday that senators for now are too divided to keep working on healthcare overhaul legislation.
But it was unclear whether White House admonishments would impact Capitol Hill, where Republicans signaled it was time to move on to other issues.
The healthcare Bill is being written behind closed doors, making the public and many legislators apprehensive.
Trump seeks to revamp healthcare plan to replace Obamacare which has been criticised by the Republics as a government overreach.
Recently some Republicans have expressed concern about the party’s current strategy of voting for a repeal without having a consensus replacement plan ready.
Ryan will face an election in January, when all members of the new house, both majority Republicans and minority Democrats, vote on a new speaker.
Trump, who will take office on January 20, will also look to fill other top administration jobs in the coming weeks.
Senate Judiciary Committee chairman Chuck Grassley, a Republican, said in a statement that an initial review of the material showed most of it was marked unclassified and urged the FBI to make as much of it public as possible.