Cut zucchini in half lengthwise; cut each half cross-wise into 1/4-inch-thick slices. Set aside.
Brown ground beef and garlic in a large Dutch oven, stirring until beef crumbles; drain. Stir in pepper, salt, broth, Italian-style stewed tomatoes, and sliced carrots; bring to a boil. Cover, reduce heat, and simmer 10 minutes.
Stir zucchini slices and beans into mixture; cover and cook, stirring occasionally, 4 to 5 minutes or until zucchini is crisp-tender. Remove from heat; stir in torn spinach.