All told, 420 members - 230 Democrats and 190 Republicans - voted for the resolution.
Current law states that the Mueller Report must be turned over to the attorney general, in this case the Trump-appointed AG, William Barr.
It called for any report Mueller sends to Barr to be released publicly - except for any classified material.
Most Republicans sided with Democrats on the resolution, though they called it a waste of time.
The measure faces an uncertain future in the Republican-led Senate.
In introducing the resolution, Nadler and five other Democratic committee chairs said "the public is clearly served by transparency with respect to any investigation that could implicate or exonerate the president and his campaign". Those regulations require only that the report explain decisions to pursue or to decline prosecutions, which could range from a bullet-point list to a report running hundreds of pages.
When the investigation concludes, Mueller is required by regulation to submit a confidential report to Attorney General William Barr. Four members of Congress voted present - Republicans Justin Amash of Michigan, Matt Gaetz of Florida, Paul Gosar of Arizona and Thomas Massie of Kentucky.
But the effort did not move forward after Republican Sen.
Manafort has been sentenced to about 7 1/2 years in federal prison after he was convicted in two cases that came out of the Mueller investigation, and shortly after he learned his fate the Manhattan district attorney indicted him on 16 pardon-proof counts related to real estate fraud. Mueller has not indicated when he will complete the report, though an announcement on Thursday about the coming departure of a senior prosecutor on his team stoked speculation that it could be soon.
"My goal will be to provide as much transparency as I can consistent with the law", he said in his opening statement, saying he believes "it is very important that the public and Congress be informed of the results of the special counsel's work".
But Democrats have said they are unsatisfied with Barr's answers and want a stronger commitment to releasing the full report, along with interview transcripts and other underlying evidence. But both McConnell and the chairman of the Senate Judiciary Committee, South Carolina Sen.
"This resolution is critical because of the many questions and criticisms of the investigation raised by the president and his administration", said House Judiciary Committee Chairman Jerrold Nadler. The letter asked for a public release of the Mueller report and that the attorney general provide to Congress any information in the report that can not be publicly released by law.