@Len Not sure what my running style is; by "very slowly" I mean I just do a slow jog that doesn't overexert me; physically I'm in excellent condition but I'm carrying about 10 extra pounds right now that I'd love to get rid of, hence the running. My main form of exercise til now has been cycling; I'm mostly a point-a-to-point-b utilitarian cyclist (I live in D.C., where a bike is just the easiest way to get around, so I'm on my bike every day) though I've done a few centuries within the past couple of years, which is to say that my legs/calves are put to work regularly. So I'm in very good shape and there's no real reason that I can see for this calf pain. I'll be more conscious of running on my heels and see if that helps any.
I'll also try not stretching before my next run. (Interesting that both of you say not to stretch beforehand; I always thought that was a given.)