Add papaya (4 cups, peeled, seeded, cubed), 2 cups of the cold water, and granulated sugar to a blender; blend on high until very smooth, 30–45 seconds.
Pour into a 2-quart pitcher and stir in the remaining 2 cups until evenly diluted. If you prefer a finer texture, pass the mixture through a fine-mesh strainer into the pitcher.
Stir in the lime juice and taste to balance sweetness and brightness. Chill until cold, 1–2 hours, or proceed to serve.
Fill glasses with ice and pour the agua de papaya over the ice. Serve immediately.