Bring water and rice to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until rice is tender and liquid has been absorbed, about 20 minutes.

Combine rice, coconut milk, sugar, and salt in a large saucepan; bring to a boil. Reduce heat and simmer until pudding is thick and creamy, 18 to 20 minutes. Stir milk and eggs into pudding, stirring constantly to keep pudding from sticking, at least 2 minutes.

Remove pudding from heat and stir in butter, vanilla, cinnamon, and nutmeg until thoroughly mixed. Fold peaches and raisins into pudding; serve warm.