And as we are at it, most nuts are not nuts either.
My vissal is, never refrigerate ripe or green tomatoes, hang them like chiles.
“Vissal”?
Google is not being useful. Weird auto-incorrect for “advice”? Word I just don’t know?
As to the ripening after picking thing, it has to do with ethylene production in the fruit. Fruits that ripen after picking are called climacteric, and include apples, melons, bananas, apricots, and tomatoes as well as most stone fruits. Non-climacteric fruits on the other hand include citrus fruits, grapes, and strawberries. However, non-climacteric melons and apricots do exist.
I’m an apple fanatic and make pilgrimages to the orchards every fall. Apples may be classified as climacteric, but I have never seen any apple’s quality improve after picking. Left out at room temperature, they just get soft and lose flavor. Once picked my experience has shown me the fridge is the place to store apples if they’re going to go uneaten for over a few days. U.S. supermarket apples have usually been off of the tree for weeks in the fall and months in the summer and spring and definitely need refrigeration.
I also have never seen a watermelon to improve in quality after picking.
What? Don’t all chefs consider cucumbers and squash to be berries? 
The Brandywine on the vine is in the vissel by the mistle.
The grape tomato by potatoes has the fresh taste that is true…
What varieties?
I think in apples this is at least partly variety dependent; though I don’t grow apples, and don’t have a lot of knowledge about them.
It can certainly make a difference for some other things – butternut squash improve in storage for as much as several months, for instance, but acorn squash doesn’t, and many acorn strains won’t keep that long.
I have taken many late season, vine picked, green tomatoes - wrapped them each, in newspaper, and stored them together in a cardboard box in the cold garage overseason. A riper tomato they become midwinter.
Actually, i just remember lining and layering my fall picked green tomatoes in a cardboard box, with newspaper cover, per the experience and knowledge, of an old timer I knew… they last a long time and ripen slowly in overwinter lager.
“The Mediterranean long shelf-life (LSL) tomatoes are a group of landraces with a fruit remaining sound up to 6–12 months after harvest. Most have been selected under semi-arid Mediterranean summer conditions with poor irrigation or rain-fed and thus, are drought tolerant.”