Is it possible to render other materials besides volumes using a procidural texture to define "density"? In other words, is it possible to render an image of a cube like this:



But make the "spheres" refractive as opposed to scattering volumes?

When I attempted with this (I did not expect it to succeed): 



I got this:



As you can see, there are no surfaces inside the cube.

Is it possible to render a procedural texture in 3D so that black parts bend or reflect light?
BSDF surface shaders only works on 'real surfaces', i.e. faces that are part of a mesh.  Surfaces inside a volume created by change in density will not be treated or rendered with a BSDF surface shader.

This image might be able to show you what I mean. The geometry has a volume shader and a glossy shader, driven by a checkered texture. Notice the shiny material is correctly applied only to the outside surface of the mesh. Interior surface boundaries remain non-glossy.