Is It Possible To Ripen An Unripe Watermelon After Buying It? - Tasting Table
Briefly

Watermelon is a non-climacteric fruit, meaning it does not continue to ripen after being picked. Once harvested, its quality and sweetness are fixed, and it does not react to ethylene gas, which influences ripening in climacteric fruits. Therefore, selecting a ripe watermelon at the store is crucial since it won’t improve once removed from the vine. Signs of ripeness include a hollow sound when tapped, yellow spots, and webbing on the skin, which indicate sweetness. Picking a ripe watermelon ensures the best flavor and tenderness at home.
The watermelon won't get any better than it already is, so it's imperative to pick out the best, ripest one when you're at the grocery store.
Once it's off the vine, this fruit's fate is sealed. No ripening. No softening.
Watermelon falls squarely into the non-climacteric category, meaning that after it's picked during its peak ripeness, it no longer produces ethylene gas.
Changes in sugar content, flavor, and tenderness still occur with watermelon, but in a much more subtle manner than the drastic change in climacteric fruits.
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