Re: What is wrong with a universe with zero energy density?

General Relativity (GR) does permit hypothetical universes with no matter
or radiation density.  Indeed, one of the first proposed solutions to the
GR field equations was the de Sitter Universe, which has space but no
matter/radiation, and "expands" due to the existence of a cosmological
constant.  In the future, as our Universe expands and the matter and other
forms of energy are diluted, so it would become more and more like a de
Sitter Universe.

The modern viewpoint tends to regard a cosmological constant, if it exists
(as seems likely from observations), as probably being due to an energy
density of the vacuum, and in that sense there is not zero energy-density.

Another early cosmological model, perhaps closer to what you are imagining,
was the Milne model, which postulates no matter, radiation or cosmological
constant.  It therefore describes a universe with no gravitational forces
at all, and exists in the Minkowski space-time of Special Relativity.  Of
course, although this model is not forbidden, our observations of
gravitating matter/energy means the Milne model is not a viable description
of our Universe.

The requirement of the ultimate unification of GR and quantum mechanics is
rather independent of these considerations, and can be thought about in a
number of different ways.  In particular, it would seem necessary to find
some quantum theory of gravity if we are to understand the singularities
which afflict current theory when applied to Black Holes or the Big Bang. 
For example, if we extrapolate our conventional theory back to the Planck
time (about 10 to the power -44 of a second) then quantum mechanics
predicts the spontaneous creation of virtual particles which would
themselves be black holes.  Exactly how nature brings about this
unification is yet to be found, but theoretical problems of this sort
suggest it must occur somehow!