Heat a 12-inch non-stick skillet over low heat. Mix tomatoes, 1/2 teaspoon oregano, 1/4 teaspoon salt, and ground black pepper together in a small bowl. Stir in feta cheese.

Whisk eggs together in a medium bowl. Stir in spinach, remaining 1/2 teaspoon oregano, remaining 1/4 teaspoon salt, and ground black pepper.

Add olive oil to the skillet and increase heat to medium-high. Heat until oil is shimmering. Pour in egg mixture. Push set eggs toward the center with a plastic or wooden spatula, tilting the skillet to allow uncooked egg to flow onto the empty portions. Continue cooking until eggs are set with no visible liquid remaining, about 3 minutes.

Reduce heat to low. Spoon tomato mixture over half of the omelet. Fold untopped half over filling with a spatula. Slide omelet onto a cutting board. Let stand for 1 to 2 minutes for filling to warm.

Cut the omelet into 4 wedges and serve immediately.