Heat oil in a stockpot over medium heat; cook and stir red onion, red bell pepper, garlic, and ginger until tender, about 5 minutes.
Mix watermelon, tomatoes, lime juice, curry powder, salt, and sugar into onion mixture; bring to a boil. Reduce heat to medium-low and simmer, uncovered, until watermelon and tomatoes are slightly broken down, about 10 minutes.
Fill 4 clean jars with chutney, seal, and turn upside down to mix for 5 minutes. Turn right-side up and refrigerate.