According to this wikipedia article, Aristotle writes the following in his book Physics :
In a void, no one could say why a thing once set in motion should stop anywhere; for why should it stop here rather than here? So that a thing will either be at rest or must be moved ad infinitum, unless something more powerful gets in its way.
To me, this sounds very much like Newton's first law of motion. It doesn't include the word "uniform" motion, of course.
My comment turned too long, and so I'm posting it as an answer.
I would like to reassure you that, yes, your reading is reasonable and it does seem like Aristotle could have arrived at the law of inertia that way. However, note that it could also have been an exponential decay that never reaches zero, say. Of course, later on, Aristotle asserts
For if the body Z be as much thinner than D as E exceeds H, A, if it moves through Z, will traverse it in a time inverse to the speed of the movement, i.e. in a time equal to H.
emphasis mine, and then yes, this is law of inertia, because there is no diminishing in speed.
One small problem is that, after your quotation and before mine, Aristotle was making an assertion that for constant distance, the time taken to traverse is in direct proportionality to some measure of "density" of the medium. This is tolerable for friction-like constant deceleration, but is already wrong for constant viscosity, and surely worse for quadratic air drag.
Aristotle's thoughts on the matter is thus quite muddied, with some wrongness mixed in with the correct stuff.
Do also note that the entire thought exercise that Aristotle is giving, is to prove that vacuum does not exist, and thus law of inertia would never apply. That, then is the historical impact that is propagated down through the ages.
So, yeah, Aristotle might have accidentally said the correct thing in the midst of doing something very wrong. This should not change our perception of his influence on history any bit.
There's a couple of things wrong here.
First, as someone has already pointed out, Aristotle was not aware of Galileo or his ideas at all because Aristotle lived about 2000 years prior to Galileo.
Second, Galileo's law of inertia is not the same as Newton's first law. Newton certainly gives credit to Galileo for the idea, but Newton refined it further. Galileo's (limited) principle of inertia only applies to horizontal motion since he recognized that there was no force or resistance to motion on a horizontal plane; however, Galileo also believed that this horizontal motion was actually just circular inertia in view of the relatively large radius of the Earth. In other words, Galileo saw horizontal inertia as just an approximation of circular inertia. Galileo also did not have a force concept for gravity that acted on falling bodies; instead, the object's fall was its natural tendency, which is countered by an opposite force equivalent to the object's weight.
To contrast this with Newton's 1st law, Newton broadened Galileo's observations by stating that inertia was rectilinear until a force acted upon it, thus inertia would continue horizontally, vertically, or some combination thereof, but the force of gravity would affect the vertical component of its motion.
Third, Aristotle argued against the existence of the void, so to say that he was "aware" of [Newton's] law of inertia is strange because the point Aristotle is trying to make is that the void, and motions within it, are absurd. As Aristotle would argue: since there's no resistance to the object's motion, then the motion must be infinitely fast in the void (i.e., "...it moves through the void with a speed beyond any ratio"), but this is absurd (i.e., "...this imposisble result will follow: it will be found to traverse a certain distance, whether this be full or void, in an equal time..."); therefore, voids do not exist. So it's not that Aristotle is recognizing inertia, it's that he's insisting that a medium must exist because the motion would be infinitely fast. Aristotle has no concept of acceleration, so motion in a medium is constant and immediately attained; Aristotle is not saying the object is accelerating to infinite speed. Also, the Peripatetics (an Aristotelian school of thought) argued that the medium must exist since it provided the motive power for motion.
Of course, Aristotle's argument against the void is severely flawed, and Galileo will specifically criticize and mock Aristotle's argument on this subject in several of his books (De motu antiquiora, Two New Sciences, to name a few). It's also interesting to note that, in Galileo's early studies, he believed that a void could exist, but an object's fall in the void would still approach some terminal velocity. He would later change this view with his studies on acceleration in which he believes that terminal velocity occurs because of air resistance.
