Why is there no permittivity-type constant for gravitation?

Permittivity $\varepsilon$ is what characterizes the amount of polarization $\mathbf{P}$ which occurs when an external electric field $\mathbf{E}$ is applied to a certain dielectric medium. The relation of the three quantities is given by

$$\mathbf{P}=\varepsilon\mathbf{E},$$

where permittivity can also be a (rank-two) tensor: this is the case in an anisotropic material.

But what does it mean for a medium to be polarized? It means that there are electric dipoles, that units of both negative and positive charge exist. But this already gives us an answer to the original question:

There are no opposite charges in gravitation, there is only one kind, namely mass, which can only be positive. Therefore there are no dipoles and no concept of polarizability. Thus, there is also no permittivity in gravitation.

This is speculation, but I would guess that it's for the same reason gravity never repels, only attracts. The electric permittivity tells you how the electrons in a material rearrange themselves to oppose an applied field. I'm not sure what the analogue for gravity would be there.