Break up the tomatoes and their juices in a medium bowl with your hands or a potato masher. Add the oil, garlic, ¾ teaspoon salt, and ½ teaspoon pepper. In a separate medium bowl, combine the uncooked rigatoni, onion, and mushrooms.
Spread a third of the tomato mixture in the bottom of a 4- to 6-quart slow cooker. Top with half the pasta mixture, half the spinach, a third of the tomato mixture, half the ricotta, and half the mozzarella; repeat. Cover and cook on high until the pasta is tender, 3 1/2 to 4 hours.