Could gravity be an emergent property of nature?

Despite all I wrote in my other answer, there's a very interesting attempt by Xiao-Gang Wen to come up with emergent models of gravity starting from quantum lattice models with no gravity, and only nearest neighbor interactions. His work can be found at gr-qc/0606100 and arXiv:0907.1203. He managed to show that quasiparticles with no energy gap and a helicity of $\pm 2$ can emerge without being accompanied by helicity $\pm 1$ or $0$ quasiparticles. Whether or not this model can be considered a model of gravity though is another matter.

I'm not an expert in gravity, however, this is what I know.

There's a hypothesis about gravity being an entropic property. The paper from Verlinde is available at arXiv. That said, I would be surprised for this to be true. The reason is simple. As you probably know, entropy is an emergent property out of statistical probability. If you have non-interacting, adimensional particles into one half of a box, with the other half empty and separated by a valve, it's probability, thus entropy, that drives the transformation. If you look at it from the energetic point of view, the energy is exactly the same before and after the transformation. This works nicely for statistical distribution, but when you have to explain why things are attracted to each other statistically, it's much harder. From the probabilistic point of view, it would be the opposite: the more degrees of freedom your particles have, the more entropy they have. A clump has less degrees of freedom, hence has less entropy, meaning that, in a closed system, the existence of gravity is baffling. This is out of my speculation, and I think I am wrong. The paper seems to be a pleasure to read, but I haven't had the chance to go through it.

You might want to look up the Weinberg-Witten theorem which shows that's not possible given certain assumptions. If the original model from which quantum gravity is supposed to emerge is an ordinary Poincaré covariant quantum field theory over flat nondynamical Minkowski space, they showed it's not possible for massless helicity $\pm 2$ particles to emerge. As a theory of quantum gravity ought to contain gravitons, this appears to rule out such models. Of course, these assumptions are questionable. For instance, the theory from which gravity emerges might not be a quantum field theory. This is the case for superstring theory.

Another possibility might be the "fundamental" model isn't Lorentz covariant. However, we still need the low energy effective theory to be approximately Lorentz covariant. In typical condensed matter analog models, different quasiparticles couple to different metrics, and there is no universality to the gravitational couplings, or the speed of light. Unless all the quasiparticles co-emerge together, I don't see any way around this problem.

It might be a bit hard to come up with the positive energy theorem in an emergent theory of gravity. The positive energy theorem states that the ADM energy of an asymptotically flat spacetime always has to be nonnegative. In an emergent theory, the ADM energy could just as easily be negative for some states. To see this, note first that the ADM energy can be defined locally as the limit as we go to spatial infinity of a locally defined integral over an enclosing spatial surface with one spatial and one time codimension. If we assume the "fundamental" theory is local, this means the now emergent ADM flux also has to be defined locally in terms of the more fundamental fields. As the enclosing boundary becomes larger and larger, its extrinsic curvature becomes closer and closer to zero. If we have a positive ADM flux passing through a plane — as defined with respect to a choice of normal vector orientation — a reflection by the plane will give us another state where this ADM flux is now negative. So, we can certainly imagine performing some sort of approximate reflection about the enclosing submanifold on a local patchwork basis, at least for the regions at or around the enclosing surface. We then need to find an interpolation of the resulting state far into the interior, which of course, might not look anything like a reflection at all. But if the fundamental theory also satisfies local independence, that ought to be possible. But the end result of all this construction is a state with negative emergent ADM energy. I know this argument is very handwavy and nonrigorous, but it sounds plausible. But there might be some loopholes. For instance, the fundamental theory might be local, but the emergent large scale excitations — and hence emergent spacetime — might be delocalized with respect to the underlying background spacetime. Or the underlying fundamental theory might be inherently nonlocal.