Unlikely. The strong primary rainbow on earth happens because the sun location and the water locations are limited. If you tried to make a sky full of rainbows, it would just smear together into white.
Ice crystals commonly form in earths atmosphere. Like diamonds, and unlike water droplets, these disperse light in different directions depending on the orientation of the crystal. Because in most situations there is no orientation to the crystals, all colors go in all directions, and the overall impression is white. This is the same thing that happens in foams and snow. Further, we see a rainbow because there is little rain between us and the edge of the rain area. If the rain were everywhere in the sky, the light would rebound around and you'd get the white of clouds.
There is a type of ice rainbow that can happen when a type of hexagonal ice forms, and then orients itself face up as it falls. This directionality allows them to disperse in a particular direction and such ice rainbows are possible. Sky diamonds would need some sort of orientation mechanism to do the same thing.
All told, you could possibly get diamond rainbows in the same way you get ice rainbows on earth. But a sky full of diamonds would probably look white (assuming white light), not spectacularly colorful.