American Public Would Blame GOP for Failure to Raise Debt Limit
If recent polling holds true, the American public would blame the Republicans in Congress should the the government fail to raise the debt limit. This makes sense given that Republicans have failed to seriously discuss any compromises put forward.
The Huffington Post has a good breakdown of the numbers (emphasis added):

When asked to whom they would assign responsibility should the limit not be raised, 48 percent of respondents chose the congressional Republicans; 34 percent said they would blame the Obama administration.
These results echo a Pew Research Center poll conducted from June 16-19 which asked the same question as the Quinnipiac poll. According to that survey, 42 percent of respondents said they would blame Republicans in Congress if the debt ceiling were not raised; 33 percent chose the Obama Administration.


