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Because the batter never reached first base and was the final out of the inning, no run may score on a play involving this batter. Therefore, R3's run is nullified. The umpire can declare the batter-runner out when he believes, in layman's terms here, that the B-R no longer made any effort to continue to first base. Some may call this abandonment, but the J/R manual refers to it as desertion. I suppose that's splitting hairs, but to make it easier to understand: did the B-R give up his advance to first and not reach it? If yes, he's out--final out, no runs score.
As far as which rule covers this run being nullified, refer to 4.09(b) for more information.
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