Prepare the grill for direct cooking over medium heat.
In a food processor puree the blackberries with the water. Add sugar to taste.
In a small saucepan over medium heat, melt the butter, then add the sugar and stir to dissolve. Remove the saucepan from the heat. Brush the peach halves all over with the butter mixture.
Grill the peach halves over direct medium heat, with the lid closed as much as possible, until they are browned in spots and warm throughout, 8 to 10 minutes, turning every 3 minutes or so. Remove the peaches from the grill and serve warm with the blackberry sauce.