As we come to the end of the peach season, it is especially important to follow the above guidelines. End-of-season peaches will not get as soft as mid-season peaches. If you are waiting for soft peaches, stop; they won't happen - you'll get mealy peaches instead. Leave the peaches out for 1-2 days and then eat.
Whole watermelon will keep in the refrigerator for 4-5 days. Cut watermelon pieces will keep in the refrigerator for 5-6 days. Whole watermelon will keep unrefrigerated for 1 day.
Because of the excessive rain this year, melons will not keep the way they have in prior years. Also remember that melons soak up a lot of water, so since this season has been rainy, they will have more water, aka, the flavor will not be as intense and the flesh will not be as firm as drier seasons.
Basically, when in doubt, refrigerate!