The ratio is not constant on surfaces of non-zero curvature. Let $d(x)$ be the ratio of the perimeter of a circle of diameter $x$ by the diameter. If the curvature is negative then this is a strictly increasing function while if the curvature is positive it is a strictly decreasing function. It's quite easy to see that with particular models for such constant curvature spaces and directly computing. For instance, for circles on a sphere this is quite straightforward: the perimeter of the circle, as the diameter increases, reaches a maximum and then starts shrinking.
I don't know of an exact formula, so will be looking forward to other answers.