To make sauce: Mince and mash garlic clove to a paste with a pinch of salt. Transfer to a bowl. Peel 1 cucumber, then halve lengthwise and remove seeds. Coarsely grate into bowl with garlic paste. Sit in yogurt, 1/3 cup mint, and 1/2 teaspoon salt to make tsatsiki.
Peel remaining cucumber and halve lengthwise. Remove seeds. Cut crosswise into slices.
Toss sliced cucumber with watermelon, remaining 1/3 cup mint, lime juice, lime zest, red onion, and feta.
Serve fruit with tsatsiki. Sprinkle with kosher salt to taste.
Note: Tsatsiki can be made 3 hours ahead and chilled.