Nothing in the feat's description limits its action to a melee attack.
Applying "common sense" to the rules can be a dangerous thing if incorrect assumptions are smuggled in unwittingly. There is nothing in the feat that describes how you move your target. It needn't be through direct impact force. A creative DM or player could come up with half a dozen narrative reasons why a sling attack might result in a target both taking damage and ending up five feet from where they were moments ago.
It is important to remember that hit points are not simply an indication of bodily health. Not every event that lowers a creature's hit points requires bodily harm to be inflicted. Here are the two definitions offered in the Player's Handbook.
Your character's hit points define how tough your character is in combat and other dangerous situations. (PHB, 177)
Hit points represent a combination of physical and mental durability, the will to live, and luck. Creatures with more hit points are more difficult to kill. Those with fewer hit points are more fragile. (PHB, 196)
Thus, even an action that involves a bullet being flung from a sling that (mechanically) scored sufficiently high on its attack roll to match or exceed the target's Armour Class might still (narratively) be described as not actually making contact with the target. If, for example, it was such a close call that the target had to take evasive action that lowered its will to live or mental durability, then this is represented (mechanically) by a loss of hit points.