Well some picked fruit ripen because their skins continue to produce
ethylene long after they were picked. Watermelon isn't one of those
fruit, however it is sensitive to ethylene;
Unfortunately it becomes soggy instead of ripening

So don't store watermelons with fruit that emit ethylene (apple,
banana, mangos etc)
and pick a ripe one by cutting a thin sliver and looking to see how red
it is. Street sellers here do that. They cut a tiny square and if you
want to examine the melon, he'll pull out the sliver and show that to
you. It's not a health risk 'cause the sliver is heaved back in and it
seals the innards.