Bring water and brown rice to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until rice is tender and liquid has been absorbed, 45 to 50 minutes.
Cut off stem of red cabbage. Peel off leaves gently, being careful not to tear. Place individual leaves gently into a large pot of water and bring to a boil. Cook for 5 minutes; reduce heat and simmer until tender but still crunchy and red, 8 to 10 minutes more. Drain.
Heat olive oil in a large skillet over medium-high heat. Sauté onion, celery, and garlic until softened, 5 to 7 minutes. Mix in turkey and season with salt. Cook and stir until meat is browned and crumbly, 5 to 7 minutes. Add crushed tomatoes and cooked rice. Cook, stirring occasionally, until flavors combine, 10 to 15 minutes more.
Place 1 cabbage leaf on top of each tortilla. Fill with turkey-rice mixture. Fold the bottom end up and roll into a tight burrito.
Heat a grill pan over medium heat. Add burritos in batches; cook, flipping as necessary, until outsides are crunchy, about 5 minutes.