CUBE the watermelon into bite-sized pieces. Set aside.
SLICE the onion thinly, to prepare pickled onions, using mandoline for even cuts if you have one available.
COMBINE the remaining pickling ingredients in a small saucepan and bring to rapid boil, then remove from the heat. Place the sliced onions in heat-proof jar and pour the liquid over to evenly coat. The onions should soften within 15 minutes, depending on thickness.
COMBINE the red wine vinegar, lemon juice, and salt in a small bowl. Slowly whisk in the olive oil until incorporated.
DRESS the arugula lightly to taste and toss in a mixing bowl, then place it in a serving dish. Top it with the cubed watermelon, crumbled cheese, and pickled onions. Finish with any remaining vinaigrette, if desired.