Heat oil in deep 12-inch skillet over medium. Cook onion 6 minutes; add garlic 30 seconds. Add tomatoes, water, 1/2 teaspoon salt, basil, and pepper; simmer 8 minutes.
Meanwhile, mix ricotta, squeezed spinach, Parmesan, zest, half mozzarella, and remaining salt. Divide over pasta sheets, leaving narrow borders; roll from short ends.
Gently simmer sauce. Arrange 8 rolls seam-side down in one layer, in 2 rows of 4; spoon sauce over tops.
Cover tightly; simmer over low 15 to 18 minutes until pasta tender and filling 165°F. Halfway through, spoon sauce over exposed dry spots.
Scatter remaining mozzarella; cover 2 minutes until melted. Remove from heat; rest 5 minutes. Serve 2 rolls with sauce per portion.