An action is required to hurl a magic stone
The magic stone spell description states:
You touch one to three pebbles and imbue them with magic. You or someone else can make a ranged spell attack with one of the pebbles by throwing it or hurling it with a sling [...]
So the question is what does making a ranged spell attack require? The section on "Actions in Combat" (specifically the "Attack" section) states:
[...] With this action, you make one melee or ranged attack. See the "Making an Attack" section for the rules that govern attacks [...]
This shows us that making a ranged attack, which includes ranged spell attacks, requires your action. If they wanted it to require a bonus action they would have specifically said so as they do with the spiritual weapon spell:
[...] When you cast the spell, you can make a melee spell attack against a creature within 5 feet of the weapon [...]
[...] As a bonus action on your turn, you can move the weapon up to 20 feet and repeat the attack against a creature within 5 feet of it [...]
Additionally, we have confirmation of this interpretation from the lead game designer Jeremy Crawford in an unofficial ruling (a tweet):
Q. The stones that get magiced from the spell 'Magic Stone', does it take an action or a bonus action to chuck them?
A. The magic stone spell imbues magic in pebbles as a bonus action. Hurling them requires a separate action.
It might use the "Hurl A Magic Stone" action and not the Attack action
Most things that use up your action don't use one of the pre-listed actions of the PHB but use their own action. As such, it may be that hurling a magic stone does not actually use the Attack action but the "Hurl A Magic Stone action". This would impact some aspects of the spell (such as if some other feature that activates when taking the Attack action, would activate).
Another thing affected is how this spell interacts with Extra Attack. The fact that the stone can be hurled by hand or by using a sling somewhat points towards this using the Attack action, but it's going to be up to your GM whether this spell uses the Attack action, or the "Hurl A Magic Stone" action; and thus, whether or not somebody with Extra Attack can hurl multiple stones in a turn.
The Attack action does not typically make ranged spell attacks except when explicitly being allowed to do so such as with the Sun Soul Monk's Radiant Sun Bolt feature:
You gain a new attack option that you can use with the Attack action. This special attack is a ranged spell attack with a range of 30 feet [...]
The magic stone spell never mentions granting a new use for the Attack action and so, as far as I can tell, hurling a stone to gain the benefits of magic stone does not use the Attack action at all.