Hi! I don't really know, but since you didn't get a response, I will give it a shot. It probably is some kind of fungal disease, but it looks very minor. Healthy vigorous persimmon trees handle leaf spot diseases with little to no serious effects. We sometimes worry too much about every little spot and don't give plants enough credit for how good they are at surviving various problems. My big old lilac tree gets powdery mildew every year by late in the season, but it is really only a cosmetic issue and doesn't seem to damage the tree any. In the fall it drops those leaves and in the spring the new leaves start out just fine.
I would just keep an eye on it. If the leaves start curling, browning, dropping off, come back and let us know. Otherwise it will be fine.