Is there an energy density associated with a Gravitational Field?

Surely there is! The classical gravitational field has an associated energy that can be computed exactly as the energy of an electric field, and it is proportional to the square of the modulus of the field. You just have to repeat every step that brings you to the expression of the energy substituting the gravitational field at every step.

EDIT: as suggested by the comments, it is worth noting that the energy density will always be negative, in this case. This is due to the fact that the gravitational potential is always attracting. Any configuration of (at least two) masses has a negative energy because, once you put the first pieces of the configuration in place, the others are attracted by the configuration. You can also see that by expressing the energy density as $\frac{1}{2}\rho\phi$, where $\rho$ is the mass distribution and $\phi$ is the gravitational potential: $\phi$ is always negative (when taking as reference $r\to\infty$, so the interaction energy density is always negative.

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P.s.: in fact, people have written to write Maxwell's equations for the gravitational field, search things like "gravitomagnetism". But then Einstein came and changed the point of view, and his theory was more predictive and powerful. EDIT: as suggested in the comments, this postscriptum does not address the question, but is just an hint of what happens when you take the gravitational field <-> electrostatic field analogy further, and perform all steps that are done to write Maxwell's equations. But this description of gravity is incomplete, and must be replaced by General Relativity.