Melt butter in a skillet over medium heat; cook and stir onion and apple in the melted butter until onion is translucent and apple is soft, 3 to 5 minutes.
Stir wine, apple cider vinegar, red wine vinegar, brown sugar, cinnamon, and nutmeg into onion mixture; bring to a boil.
Add cabbage to onion mixture and stir to coat; season with salt and pepper. Reduce heat and simmer until cabbage has softened, about 30 minutes.