Preheat the broiler. Heat the olive oil in a medium pot over medium-high heat. Add the bell pepper, onion, garlic, cumin, salt and pepper and cook until slightly soft, about 3 minutes. Add the tomato sauce and 1/2 cup water and bring to a simmer.
Set aside about half of the tomato sauce mixture in a bowl. Add 3/4 cup more water and the rice to the pot and bring to a boil. Add the beans and their liquid, stir and reduce the heat to low. Cover and cook, undisturbed, until the rice is tender, about 15 minutes. Remove from the heat and let sit, covered, 5 more minutes.
Meanwhile, pierce the pork chops with a fork and rub with the vinegar, and salt and pepper to taste. Arrange on a foil-lined broiler pan. Pour the reserved tomato mixture over the pork chops and turn to coat. Broil, without turning, until golden brown, about 8 minutes.
Fluff the rice and beans with a fork. Season with salt and pepper. Divide the pork chops and rice and beans among plates and drizzle with the broiler-pan juices.