In a medium pan, bring the rice, water, and salt to a boil. When it starts to boil, reduce the heat to medium low and simmer covered for 15 minutes (or until water is absorbed).
Meanwhile, whisk the milk, sugar, vanilla, cinnamon, and nutmeg until sugar dissolves.
Stir the milk mixture into the rice and increase the temperature to medium.
Cook uncovered, stirring frequently, for about 35 minutes, until the rice and milk are nicely thickened and creamy like pudding.
Stir constantly the last 5 minutes or so to prevent burning. Reduce the temperature if necessary.
Remove from heat, and pour into individual servings. Cover each serving with a piece of plastic wrap placed directly on the top of the pudding to prevent a skin.
Let cool. Refrigerate if you aren't eating it right away.
When serving, top with peaches or layer the rice pudding and peaches in a parfait glass.