Fruit kept at room temperature are best eaten with a day or two.  Whole honeydew melons kept at room temperature may ripen and develop a little more flavor but are best consumed as soon as possible.

Watermelons, rockmelons and honeydew melons will keep 3 to 4 days or longer when refrigerated.  Cut surfaces of fruit are best covered to prevent them drying out or deteriorating.

Note that rockmelons do not ripen once picked, the flesh firmness reduces with time and this gives the impression of getting sweeter.