I think its likely that Obama would have replaced her with his OTL pick, Merrick Garland.
Then again, the Democrats controlled the Senate at the time, so really, he could go with someone a bit more extreme than that.
2013 saw the Republican Party lose control of some of the more opinionated members of the House, largely at the urging of Ted Cruz, in the budget fight. But 2013 was probably a fatal year for intraRepublican relations anyways. Keep in mind what happened in the year. The Republican autopsy and the attempt at immigration reform infuriated the vast majority of the party, who saw it as either ignoring the clear wishes of most Republican voters, or more cynically, as an act of abject stupidity that the party would pay for at the ballot box because they would not be rewarded by those who got amnesty (who by every indication would vote for the Democrats overwhelmingly for economic reasons even if immigration was taken off the table), and it would sap the energy of the existing voting base. The collapse of Syria also helped to drive a larger wedge between the noninterventionist and internationalist wings of the party, as Obama's false starts on the issue of chemical weapons and subsequent stand down were not met with any sort of unified or interested GOP response.
So basically, the Republican party is not going to cooperate on an Obama supreme court pick in 2013. They might have if it was very early in the year, or if some kind of deal was cut with McConnell over a pet issue of his (like a reduction in financial industry regulations, or a cut in medicaid) and he could bring over enough to break the filibuster. But my guess is that the Democrats will simply kill the filibuster in 2013, faced with intransigent opposition.