What I learned from my Health Department Food Handler's classes was that ALL fruits and vegetables, as long as they are whole, and unblemished, can be left out of the refrigerater. Once they're cut or punctured (breaking the skin of the vegetable), they need to be refrigerated.
That said, most veggies stay in my fridge, just to prolong shelf life. I leave potatoes in a dark, well-ventilated spot. Tomatoes I refrigerate. Yes, I know they loose flavor. However, my kids will eat tomatoes out-of-hand like apples if they see them. So, in the fridge; out-of-sight, out-of-mind. I also refrigerate my onions, to keep the tear factor down. Onions get used really fast around here, so they rarely, if ever, spoil.
From personal experience, fresh broccoli needs to stay refrigerated. If you leave it out, it will either start to bloom, or will discolor very rapidly.
Peace, Love, and Vegetable Rights!
Eat Meat and Save the Plants!