Add tomatoes, 1 cup water, and 3 tablespoons lemon juice; bring to boil. Arrange all chicken in single layer in pot; spoon some sauce over. Bring to boil. Reduce heat to medium-low, cover, and simmer 15 minutes. Turn chicken over, cover, and simmer until chicken is tender, about 20 minutes longer.
Meanwhile, preheat oven to 400F.
Brush large rimmed baking sheet with olive oil.
Place eggplant and remaining 4 tablespoons olive oil in large bowl; toss to coat.
Spread eggplant out on prepared baking sheet and bake until soft and brown, stirring occasionally, about 25 minutes. (Chicken and eggplant can be made 1 day ahead. Cool slightly. Refrigerate separately until cold, then cover and keep refrigerated.)
Stir eggplant and marjoram into chicken. Simmer uncovered 10 minutes to heat through and blend flavors. Season stew to taste with more lemon juice, if desired, and salt and pepper.