It would be helpful to know the age of your tree; have you used a spray of any kind on it, or on other plants around it; and have you already harvested some of the fruit from this tree (the crop seems a little light)? Did the tree look healthy during last summer?
From the photo that you provided, I have been able to rule out blue mold and freeze damage. It appears that the damage is relatively uniform on your tree. I have unsuccessfully tried to find information, or photos of persimmons with similar damage to yours.
To get proper identification of the damage on your persimmons it would be helpful if you would bring two or three affected persimmons to the Benton County Extension office, 4077 SW Research Way, Corvallis. I recommend that you call (541 766-6750) prior to going to ensure the Horticulture Agent or a Master Gardener is in the office.
To manage whatever is occurring with your tree here are some suggestions. Remove all of the damaged fruit, rake up all of the leaves and debris from under the tree and bag it, to be hauled away. Do not place this material in your compost.
Thank you for submitting this challenging question to “Ask an Expert”. We want to resolve your problem by identifying the cause of the damage and to provide a solution for future crops.