The move came after the party failed to gather enough support for the politically-charged but largely symbolic measure meant to position it for final compromise talks with President Barack Obama.
That complicates attempts to avoid the year-end fiscal cliff, the package of automatic spending cuts and tax increases that threatens to send the US economy back into recession.
In a brief statement, House of Refresentatives speaker John Boehner said the Bill "did not have sufficient support from our members to pass". At the same time he challenged Mr Obama and Senate majority leader Harry Reid to work on legislation to avert the crisis.
Emerging from a hurriedly-called evening meeting of House Republicans, congressman Steve LaTourette said Mr Boehner had told politicians: "He's going to call the president and he's going to go down and talk to him and maybe they can hammer something out."
Even if Mr Boehner's so-called Plan B legislation - drafted unilaterally by Republicans - had passed the House, the White House had threatened a veto and Senate Democrats had made plain they would sidetrack the Bill the moment it arrived from the House. Yet officials in both parties suggested the House vote would have cleared the way for a final stab at negotiations.
The White House later said President Obama was willing to continue bargaining for a cross-party solution to avert the fiscal cliff. Spokesman Jay Carney said Mr Obama intended to work with Congress, but did not mention Mr Boehner's failure to muster the votes for his Plan B.
The fiscal cliff has dominated the post-election session of Congress that now seems certain to extend well beyond Christmas. More broadly, it marks the end of a tumultuous two-year period in which dozens of conservative Republicans roared into the House demanding lower taxes, yet now find themselves two years later called on by their own leadership to raise rates on upper incomes.
Ironically, the votes were set in motion earlier in the week, after Mr Boehner and Mr Obama had significantly narrowed their differences on a compromise to avoid the fiscal cliff.
Mr Obama wants to raise taxes by about 20 billion dollars a year more than Mr Boehner. That is real money by most measures, yet such numbers are barely noticeable compared with the 2.6 trillion the government is projected to collect next year and the 3.6 trillion it is expected to spend. Despite their differences, the numbers being proposed by Mr Obama and Mr Boehner are so close and the political risks both men have taken on taxes and Social Security benefits so stark, that many consider it almost unthinkable that they would not eventually complete a deal.