Bring water with salt to a boil in a 2-quart heavy saucepan and stir in rice. Cover pan and reduce heat to low, then cook until water is absorbed, about 15 minutes.
Stir in milk and sugar and cook over very low heat, covered, until mixture resembles a thick soup, 50 minutes to 1 hour.
Whisk together egg, egg whites, vanilla, cardamom, and a pinch of salt.
Whisk about 1 cup hot rice mixture into egg mixture, then stir mixture into remaining rice.
Cook over low heat (do not let boil), whisking constantly, until an instant-read thermometer registers 170 degrees F and thick, 1 to 2 minutes.
Remove from heat and stir in raisins and cherries.
Transfer pudding to a 1-quart dish and chill, its surface covered with waxed paper, until cool but not cold, 1 to 2 hours.
Each serving contains about 192 calories and 2 grams fat.