It is hard to answer the question as asked because it presumes that people first developed the idea of air drag force that alters motion of objects on Earth, and then realized that laws of motion can be made universal by taking that into account. This is not, however, how things progressed.
People knew of air resistance, in some fashion, since prehistoric times. Aristotle already considered air to be a corporeal fluid that resists motion. It is also true that he sharply separated the laws of motion on Earth and in the Heavens by ascribing "natural" motions towards the Earth's center for the former, and circular ones for the latter. According to his idea of "violent" motion, a "force" had to be constantly applied to produce it, and this force was proportional to the resulting speed and inverse proportional to the medium resitance (at least, in modernized reconstructions of what he says, see Rovelli, Aristotle's Physics: a Physicist's Look (And Others)).
Aristotle's explanation of arrow's flight was based on a quaint idea of antiperistasis, where an arrow is resisted by air from the front, but pushed by it from behind, as it "fills the void left by the advancing tail" because "nature abhors vacuum". This idea was so quaint that he himself expressed doubts about it in Physica VIII, and it was outright mocked already in antiquity by Philoponus, who sarcastically suggested making an arrow move by waiving hands behind it. With Philoponus (or possibly already Hipparchus) originates the idea of impetus, stored energy and/or momentum, that maintains motion without application of force, see Why don't we learn Buridan's laws of motion? This was the prototype of inertia, which turned air into a source of resistance only. But the conception of force, for him and medieval theorists of impetus, remained basically Aristotelian until Leonardo and Galileo, see Hanson, Aristotle on Motion through Air.
The questioning of the fundamental difference between the Earth and the Heavens became natural after Copernicus, who displaced Earth from its place of pride at the center, and especially Kepler, but air drag played little role in it. The study of air and vacuum progressed on its own course, and we find the idea that air resistance can be disregarded when studying fundamentals of motion clearly spelled out already by Galileo. But, while he rejected the Aristotelian conception of force, he did not yet offer an alternative of his own. Descartes, Huygens and others working on collisions and pendulums contributed to its emergence in Newton's work, see How did Newton measure forces in his experiments to establish the laws of motion? So the fully modern notion of drag force, accounting for resistance in a non-Aristotelian fashion, and the idea of air being the something different between the motion on Earth and in the Heavens are not put together until book II of Newton's Principia.
Here is from Hanson:
"Regarding an object's motion through an aerial medium, Aristotle correctly noted the latter's resisting and assisting effect on the former. Aristotle's doctrine of projectile motion as founded on the assisting action of the air (itself construed as a fluid medium), was opposed in the 6th century by Philoponus... This Greek grammarian urged that the casting of a projectile imparts to it a certain energy, or impetus, which is transported from the throwing agent and somehow "transfused" into that which is thrown... Philoponus felt that it was this energy, or impetus, which maintained the arrow in motion during its trajectory long after having left the bowstring. The air only resisted its progress forward.
[...] Aristotle's medium theory and Philoponus' first intimations of an impetus theory remained in opposition to each other until the collective insights of Descartes, Galileo and Newton established the Law of Inertia beyond question. By the late 17th century, all serious controversy on the matter was virtually terminated; the air was thenceforth conceived of only as a factor of resistance to the motion of thrown or flown objects. No one continued to argue, a la Aristotle, that the aerial flow could actually assist in some manner the forward and upward flight of a projectile, or of a bird.
[...] Leonardo da Vinci was an interesting interim figure, albeit historically ineffectual. Before 1506 he had assumed that some action of the air did assist motion. After this date, however, he thought of air largely as a resisting medium, which resistance he ascribed to its "compressibility." It was this compressibility, indeed, which Leonardo identified as the cause of lift in bird flight.
[...] Galileo, whose Mechanics crushed the Aristotelian theory of the "pushing" medium, recognized a proportionality between the air's compressibility and the velocity of objects moving through it. One might hardly have expected anything else: the porpoise would ply through thick oil, or liquid mercury, or a tank of sand, much slower than through water. Galileo's work with pendulum pairs, involving precise calculations for the vacuum condition, led to his conclusion that projectile paths, and pendulums, could be described in a manner unaltered by air resistance."