Would a universe with special-relativistic gravity make sense? Coulomb's law in electrostatics in analogous with Newtonian gravity. It's pretty clear neither of these can be used in a universe that obeys special relativity. They both must be modified to avoid instantaneous communication (see this question) among other problems. For electromagnetism, we get maxwells equations. For gravity, we have the analogous gravitoelectromagnitism and gravitational waves. However, gravity also causes spacetime curvature.
But what would a "special relativistic" theory of gravity with flat spacetime look like?
Like our universe, Newtonian gravity would still be accurate for describing the solar system. Mercuries orbit would still precess due to special relativity, though maybe at a different rate, it would be the same if we placed a charged test particle in orbit around an electrostatic well at a high speed.
Like our universe, the speed of gravity would equal the speed of light. Gravitational waves would be emitted by orbiting objects and cause the orbits to shrink over time. Since all forms of energy gravitate, the waves would still exert a force on matter, which would change the distance between two mirrors, and thus the waves would still be detectable.
Like in our universe, we still would get gravitational lensing.
Kinetic energy would still contribute to a star-cluster's total gravitational field. Also, an object in a gravity well would contribute less to the gravity felt by a distant observer.
The crucial difference, however, is that "special relativistic gravity" wouldn't have gravitational time dilation. There would be no black holes.
Besides the difficulty in setting up a big-bang/cosmology, what are the fundamental problems with a universe like this? My reasoning is as follows: Suppose an object with mass m resting in a deep gravity well (i.e. the center of a very dense globular cluster) were to convert it's entire mass into a spherical pulse of light (an idealized matter-antimatter reaction), emitting m of energy to a nearby observer. Suppose it fell in from infinity. Since it gave up potential energy V to come to rest in the well, to conserve energy we must have m + E = V, where E is the light energy emitted to infinity. This necessitates gravitational redshift because E < m. Red-shifting without time dilation would mean that distant observers see each photon have less energy but the timing of any light pulses arriving is not slowed down (this is different from redshift in our universe). Although unusual, this doesn't seem to make for an obvious contradiction, it would be analogous to firing ultrarelativistic electrons out of an electrostatic well. Is there a nice argument to show such a universe couldn't exist?
 A: 
The crucial difference, however, is that "special relativistic gravity" wouldn't have gravitational time dilation.

Note that the equivalence principle implies gravitational time dilation, so if you don't have gravitational time dilation, then you have to violate the equivalence principle somehow.
But basically, what you suggest is the obvious first thing to try, and it's what Einstein tried first, ca. 1906. See Weinstein, "Einstein's Pathway to the Equivalence Principle 1905-1907," http://arxiv.org/abs/1208.5137 . She gives a translation of a lecture Einstein gave in 1933:

... I attempted to treat the law of gravity within the framework of the special theory of relativity.
Like most writers at the time, I tried to establish a field-law for gravitation, since it was no longer possible to introduce direct action at a distance, at least in any natural way, because of the abolition of the notion of absolute simultaneity.
The simplest thing was, of course, to retain the Laplacian scalar potential of gravity, and to complete the equation of Poisson in an obvious way by a term differentiated with respect to time in such a way, so that the special theory of relativity was satisfied. Also the law of motion of the mass point in a gravitational field had to be adapted to the special theory of relativity. The path here was less clearly marked out, since the inertial mass of a body could depend on the gravitational potential. In fact, this was to be expected on account of the inertia of energy.
These investigations, however, led to a result which raised my strong suspicions. According to classical mechanics, the vertical acceleration of a body in the vertical gravitational field is independent of the horizontal component of its velocity ... But according to the theory I tried, the acceleration of a falling body was not independent of its horizontal velocity, or the internal energy of the system.

This led him to the equivalence principle and its implication of gravitational time dilation, which he published in 1907.
