The success of the 1994 Republican Revolution was not a direct result of the failure of the 1993 healthcare reforms, although the reforms were part of a larger narrative that the Republicans pushed as part of the Republican Revolution.
During the first two years of his presidency Clinton faced a mounting deficit. He was reversing the tide from the Reagan/Bush years, but he was facing political and popular opposition over tax policies. During the 1988 Republican National Convention George H.W. Bush had given his famous 'read my lips' speech, in which he pledged that during a Bush presidency there would be no increase of taxes. Backed into a corner by Democratic control of both the House and the Senate however, Bush was forced to compromise with Democrats on the Omnibus Budget Reconciliation Act of 1990 in a way that forced him to raise taxes regardless. By the time the 1992 Presidential Elections took place Bush was attacked from both within his own party and from Clinton, who repeatedly attacked Bush over the remarks as proof that he wasn't trustworthy. The first two years of the Clinton presidency were marked by Republicans framing Clinton's policies as even less trustworthy than that what he had attacked Bush on during the 1992 campaign.
It should be noted that the Republican pushback against the 1993 healthcare reforms was at least partially motivated by concerns over the political gains Democrats were to gain if the reforms were successful. Republican strategist Bill Kristol, until 1993 Chief of Staff to the Vice President, wrote a now infamous memo to Republican legislators on behalf of the Project for a Republican Future, in which he warns that healthcare reform wouldn't simply be an expansion of the federal government but, if successful, would also firmly establish the Democratic party as the true protectors of the middle class, at the cost of the Republicans. In short, Republicans pushed back against the 1993 healthcare reforms not just for ideological reasons (although that definitely played a part) but also because they were worried that the Democrats would reap the electoral benefits from a successful healthcare reform.
The rhetoric of the Republican Revolution wasn't a direct push against the healthcare reform. Rather, it became part of a larger narrative in which Republicans pushed back against Clinton by depicting him not as the moderate he campaigned at as 1992, but as a stereotypical tax and spend liberal who would, if left unchecked, raise taxes across the board to pay for rampant federal government programs that would be inefficient and rife with corruption. The 1994 midterm elections were notable because the Republicans were running as a single party, under a single strategy, named the Contract for America. It was a contract signed with much fanfare six weeks ahead of the 1994 midterm election, by an overwhelming majority of Republican candidates.
So the success of the 1994 Republican Revolution campaign wasn't a direct result of backlash against the 1993 healthcare reforms, but the Republicans did effectively use the reforms as part of a larger narrative in which the Democrats in general, and Clinton in particular, would, if they were allowed to, increase the size of government to untenable levels, and raise taxes to pay for it.
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