In a word, nope. Washing produce immediately before storing it, whether the item needs to be at room temp or in the fridge, may seem like the smart route, but it actually causes the produce to spoil faster because dampness makes bacteria grow. Hold off until you're ready to cut and use the produce before washing.
While we're on the topic, an extra tip: don't cut up produce until you're ready to eat it. You may be tempted to break that big ol' watermelon down into bite-sized cubes to chill in a bowl sealed tightly with plastic wrap for convenient access, but you're doing the fruit no favors. Cutting produce ruptures its cells, causing it to deteriorate faster. (You'd perish more quickly too, if sliced repeatedly.) Stored whole, most produce will last roughly twice as long as it would chopped up.