Recent direct observation of gravitational perturbations attributed to merging black holes and merging neutron stars has reliably confirmed the existence of gravitational waves. The observed fact that the orbits of binary pulsars decay, and the assumption that the decay is due to emission of gravitational waves, is interpreted by some to imply that the energy density of the gravitational field is positive.
Here're relevant quotes from Steve Carlip:
To make gravity attractive in such a [vector-like] theory, you must require that the gravitational field has negative energy, which (apart from the obvious instabilities) would drastically disagree with binary pulsar observations
In electromagnetism, opposite charges attract, and electromagnetic waves carry positive energy. To get like "charges" to attract, you have to change a sign, and this changes the sign of the energy carried by a wave. In a vector theory of gravity, where like masses attract, a gravitational wave would have to carry off negative energy. This would mean that a binary pulsar system would increase in energy as it emitted gravitational waves, which contradicts observation.
Observations do seem to show that the energy content of the gravitational waves emitted by the orbiting pulsars is positive, but it's not obvious (to me, at least) that this requires the energy density of the gravitational field to be positive. For example, suppose the gravitational field energy density is negative, and a gravitational wave is a perturbation in which the field strength is reduced within the wave. This should give the gravitational wave net positive energy, but allow the gravitational field energy density to be negative.
This article by Guth at Cal Tech appears to contradict Carlip:
The resolution to the energy paradox lies in the subtle behavior of gravity. Although it has not been widely appreciated, Newtonian physics unambiguously implies that the energy of a gravitational field is always negative a fact which holds also in general relativity. The Newtonian argument closely parallels the derivation of the energy density of an electrostatic field, except that the answer has the opposite sign because the force law has the opposite sign: two positive masses attract, while two positive charges repel. The possibility that the negative energy of gravity balance the positive energy for the matter of the Universe was suggested as early as 1932 by Richard Tolman, although a viable mechanism for the energy transfer was not known.
The widely used WEC (weak energy condition) asserts that the energy density everywhere is non-negative in GR; but in most of the references I've read, the context is cosmological, and there is an assumption that the universe is filled with a perfect fluid having a mass density.
Is there other observational evidence that can clarify this?
Are there inescapable theoretical arguments in General Relativity that require the gravitational field to have negative energy density?
Patrick Dürr argues persuasively that gravitational waves do not carry energy.
Another perspective on energy density in the gravitational field and in gravitational waves (the two not being exactly the same!) is given by the last contribution by "Demystifier" in Physics Forums. There, it is argued that although the energy density of the gravitational field is negative, the perturbation due to a gravitational wave essentially reduces local gravitational field strength, so that the wave thereby effectively carries positive energy despite the energy density of the gravitational field being negative. This, if correct, might help resolve the otherwise apparently conflicting positions of different authorities on the subject.
Another set of possible observational evidence would be direct measurements of the Sun's gravitational field as a function of distance from the Sun. If the static gravitational field's energy density is positive (or negative), and energy is a contributing source to the gravitational field, then the gravitational field will not vary exactly as $1/r^2$; and the deviation from $1/r^2$ should tell us something.
Deviations near a black hole or neutron star would probably be easier to measure. However, the unknown distribution of dark matter is likely to make the interpretation of such measurements very uncertain.