In a fine strainer, rinse rice well and drain. In a 2- to 3-quart pan over high heat, bring 3 cups water and the rice to a boil. Reduce heat to medium-high and cook, uncovered, until most of the liquid is absorbed, about 10 minutes. Reduce heat to low, cover, and cook until rice is tender to bite, about 10 minutes longer. Scoop into a large bowl and let cool, stirring occasionally, 15 to 20 minutes.
Meanwhile, rinse cucumber, zucchini, and apple. Trim off and discard ends of cucumber and zucchini; cut vegetables into 1/2-inch cubes. Stem and core apple; cut into 1/2-inch cubes. Add cucumber, zucchini, apple, and onion to cool rice.
Pour any excess liquid off yogurt. In a bowl, mix yogurt, ginger, and 1/2 teaspoon salt. Add to rice mixture and stir gently to coat.
Set a 6- to 8-inch frying pan over medium-high heat; when hot, add oil and chili flakes and stir just until chili begins to darken slightly, 15 to 30 seconds. Add pepper. Pour over rice mixture and stir gently to mix, adding more salt to taste. Spoon into a serving bowl and garnish with shredded carrots.