For the pickled watermelon rind: Slice off each end of the watermelon and discard. Stand the watermelon up and begin trimming the thin green layer off, leaving the white pith layer on the watermelon. Depending on the size, slice the watermelon in half widthwise so it is easier to work with.
Trim the white rind off in the same way the green rind was removed. When all the white rind has been removed, trim each piece into a 1-by-6-inch plank. (This measurement doesn't have to be exact, you are just trying to create pieces of relatively the same size and thickness.) Trim any remaining pink parts off. Wedge as many planks as you can into a 32-ounce wide-mouthed glass jar with lid. Discard the rest.
Combine the cider vinegar, 1 cup water, sugar, salt and aromatics in a saucepot over high heat and bring to a boil. Whisk to dissolve the sugar, then pour over the rinds while hot. Allow to cool to room temperature before refrigerating. Refrigerate overnight before using. The rinds will keep up to 1 month in the refrigerator.
For the ouzo spritz: Fill 2 lowball glasses with ice. Place the ouzo, lemon juice, egg whites and pickling liquid in a cocktail shaker without ice. Vigorously shake to froth the egg whites, then strain into the glasses. Top with seltzer and garnish with a spear of pickled watermelon rind and a sprig of mint.