Where's the study itself? A pre-publication version was posted at http://www.atmos.umd.edu/~ekalnay/pubs/handy-paper-for-submission-2.pdf where I looked at it briefly, but now I'm getting that as a broken link (because of too much traffic or has it been taken down?); other than that, I suppose you'll have to wait for an upcoming issue of Ecological Economics, where it has reportedly been accepted for publication.
The essence of the research it seems is four equations, including a model of "elites" and "commoners" as predators and prey. For more, see this David Appell post. Does the study offer any valuable insights? Who knows (at least for now, given that it seemingly cannot be accessed)? But one thing that's clear is that weighing its credibility was not a big factor in its rapid dissemination. Another is that it found resonance on many points of the ideological spectrum. And here's a bit of equally clueless backlash: "NASA Goes from 'Muslim Outreach' to Leftwing Activism."
UPDATE 3/20, 12:10pm: And here's the (or a version of the) pre-publication paper (PDF): "A Minimal Model for Human and Nature Interaction." As for the NASA role, it says "This work was partially funded through NASA/GSFC grant NNX12AD03A, known as 'Collaborative Earth System Science Research Between NASA/GSFC and UMCP.'" Searching for info on that grant, one gets the following (though no breakout as to how much of the total went to this particular paper):
Researcher: Antonio Busalacchi Grant Sponsor: NASA-Goddard Space Flight Center
Amount: $334,835 in additional funding, bringing the total award to $9,344,034
Grant Study: "Collaborative Earth System Science Research between NASA/GSFC and UMCP"
UPDATE 3/20, 12:33pm: OK, I've read it, which didn't take long. This is, as Appell wrote, a "toy model." That is, it's an extremely simplified approach that abstracts away, among other things, any government policies, the existence of different regions, differences in types of natural resources, most socioeconomic distinctions (there are just "commoners" and "elites," and the elites keep on consuming while the commoners sink into famine). I hope that NASA did not spend much money on this, and if it did, the awful "How NASA Thinks Society Will Collapse" media coverage begins to seem like a fitting punishment.
UPDATE 3/21: NASA's statement, released yesterday, referring to a version of the paper with a different title than the one I mentioned:
NASA Statement on Sustainability Study
UPDATE 3/22: About time to put this sordid tale from the Age of Misinformation behind us, but first a final pair of links. One is intelligent: "NASA Doesn't Think the World Is Ending and Journalists Should Stop Implying That." The other is, well, the final punchline, and includes this gem: "One would think that a scientific paper predicting the imminent end of civilization, backed by the imprimatur of no less an authority than NASA’s Goddard Space Flight Center, would generate considerable interest, not to mention news coverage. Oddly, though, it hasn’t, at least not in North America."