My father, z”l, used to work in the produce department of a supermarket. I learned a lot from him. I would suggest you look at how a supermarket stores their fruits and veges. There are sections that are with refrigeration and even a watering system. There are other sections that are not. I think any fruit or vege can be kept without refrigeration until it is ready (or almost just ready) to eat. Then if you want to keep them from spoiling, refrigerate. Exception: Potatoes and onions. Like OOM says, if you buy fruit that is not yet ripe, putting it into a paper bag will speed the ripening process, but once it is ripe, put it in the fridge to keep it a little longer if you’re not yet ready to eat it (like you need it for Shabbos, YT, etc). I put all cukes and tomatoes into the fridge unless they’re green. Nothing like a cold cuke on a hot summer day!!!
Also, with melons, like cantaloupe and watermelon, the longer they are kept unrefrigerated, the more they ripen and the more the sugars inside become concentrated, leading to a sweeter tasting melon. Once you cut it, into the fridge it goes.