The equations are straightforward and well-known. All you need is a plausible scenario for the planet to be non-rocky and non-metallic.

The Moons of Jupiter and Saturn appear to be very ice rich. In principle, a 1g ice/waterworld could exist as a moon around a Superjovian.

g ~ M/r^2

surface gravity is proportional to central mass divided by radius squared

M ~ rho.r^3

mass is proportional to density times radius cubed

hence g ~ rho.r

Earth has a density rho of 5.9. a water/ice planet would have minimum rho about 1.5 (gotta have some other junk mixed in with it), therefore r would need to be 5.5/1.5 times Earth's to get 1g at surface. 3.67 times 6371km is about 21,000 km - something like the size of the Great Red Spot on Jupiter.

Can't find a handy reference to the density of diamond but I'd guess it's about half that of the averge Earth, so a diamond/carbon planet would be 12-13,000 km in radius to give 1g.