A grappled creature can use its action to escape.
It says an action, so it requires an action. If it could be done as one of several attacks, it would say so, just like it says for making a grapple.
If you're the DM, you can houserule this, of course, but think carefully before doing so, because allowing one attack of Extra Attack to escape grapples vastly favours the players. There are some monsters that are only dangerous because of grappling, so making it easier to escape a grapple is fairly crippling for them. Also, due to the way Multiattack works, this will only make it easier for players to escape monster grapples, not for monsters to escape player grapples.
Officially, this is one of the handful of things the game says that a lone typical creature just can't do with a typical weapon…
As per this excellent answer, in official material there are ways for the typical creature to sort of take a free action to sheathe a weapon, but these are largely workarounds for an option never officially never provided beyond the gnome quickrazor (Races of Stone 154, 155) (45 gp; 1 lb.).
(I've always assumed that the inability to sheathe a weapon as a free action was to prevent overuse of the skill Iaijutsu Focus (Oriental Adventures 58-9, 81-2), which says that an "attack [against] a flat-footed opponent immediately after drawing a melee weapon… can deal [up to +9d6 (or, at epic levels, even more)] extra damage, based on the result of an Iaijutsu Focus check" (81), therefore requiring a newly unsheathed weapon to be sheathed again before the skill can be used a second time with that same weapon. Shockingly, a warrior doesn't usually want to drop her weapon after making that one attack (or, as some argue, her whole round's attacks) and use a different weapon; she'd rather find a way to sheathe and redraw the same weapon. The game knows this, and, even at the bitter end, sheathing a weapon as a free action remained one of the few officially-almost-impossible D&D 3.5 tasks.)
...But third party says, "Meh. Whatever."
With third party material available, this becomes much easier. For example, Paradigm's Arcanis Player's Guide (2004) includes this Open-Gaming-Content feat:
Quick Sheathe [Fighter, General]
You have learned to put items away or snatch them up off the ground in the blink of an eye.
Prerequisites: Dex 13, Quick Draw or Quick Hands.
Benefit: Once per round on your turn, you can sheathe or holster your weapon, or put away a held item, as a free action that does not provoke attacks of opportunity.
Once per round on your turn, you may attempt a Reflex save (DC 15) to retrieve a weapon or item from the ground without provoking an attack of opportunity, as long as the item to be retrieved is in your square. If you fail the save, you still retrieve the item, but you suffer attacks of opportunity as normal. Picking an object up still requires a move action.
Normal: Putting away a weapon or object is normally a move-equivalent action and provokes attacks of opportunity. (187
(The Arcanis Player's Guide also describes the feat Quick Hands (186), but the feat Quick Draw (Player's Handbook 98) is typically a better choice for meeting this feat's prerequisite.)
I am certain other third party texts provide similar options, but this is the first I happened upon.
Best Answer
The only indication we have within the rules that dropping things might be free is that dropping an object is not included in the (fairly extensive) list of example object interactions.
However, Jeremy Crawford has stated in an unofficial tweet from April 2015 that dropping something you're holding is not intended to require any action.