I don't know about moon sized but there are solar systems out there where carbon is more common than silicon. In such a system if you had a terrestrial planet then you're likely get diamonds instead of quartz being the most common mineral in the crust. You also might possibly get diamonds in an octagonal crystalline form which are theorized to be far stronger than the diamonds we have here on Earth.

The density of diamond is about 3.5 g/cm^3. The Earth's moon has a density of around 3.3 g/cm^3, so if you replaced it with a diamond of about the same size, it actually wouldn't be all that different in terms of gravity. Solar eclipses would be pretty wild though.

Was more thinking about whether there are any interesting gravitational effects on the internal chemistry of a giant crystal. Say it is one single crystal, how would its centre differ from the surface?

Earth, Moon, or Big Fine Diamond, the gravity at the surface is 1 unit (for whatever normalised radius, density to get a 1 works) while the gravity at the centre is 0.

It's certainly polycrystalline rather than a giant single-crystal. It would contain lots of every other element that's soluble in it, to its limit of solubility, and whatever's insoluble or over that limit would have to form different mineral inclusions at grain boundaries.