No, he did not. Not in the first (1687), not in the second (1713), and not in the third (1726) edition, below I quote from Cajori's edition of Motte's translation of the latter. Since he was unable to find the trajectory analytically he tried to approximate it by pre-chosen curves. In the process he
apparently messed up with the semi-circles in the first edition of Principia, which Johann (=John) Bernoulli pointed out in 1712. Newton fixed the error in the next year's second edition, without mentioning Bernoulli. The problem was offered as a challenge in 1717, when Bernoulli solved it (in a parametric form) as a particular case of the problem with resistance proportional to an arbitrary power of speed.
Newton did find the trajectory for the linear resistance law, suggested by Galileo, in Section 1 of Book II. But he argued that the square law is correct for rare media like the air, and even that it can be derived from his laws of motion! Section 2 examines this case, first without gravity, and then for the fall in a straight line. But the closest Newton comes to the general trajectory is when he studies the inverse problem under Proposition X of Book II:
"Suppose the uniform force of gravity to tend directly to the plane of the
horizon, and the resistance to be as the product of the density of the medium
and the square of the velocity: it is proposed to find the density of
the medium in each place, which shall make the body move in any given
curved line, the velocity of the body, and the resistance of the medium in
each place."
He considers four types of trajectories - semi-circle, parabola, hyperbola with oblique and vertical asymptote, and generalized hyperbola with similar asymptotes - and reasons what kind of medium distribution would produce them. His heuristic conclusion for the uniformly dense medium is this:
...the line which a projectile describes in an uniformly resisting medium approaches nearer to these hyperbolas than to a parabola. That line is certainly of the hyperbolic kind, but about the vertex it is more distant
from the asymptotes, and in the parts remote from the vertex draws nearer to them than these hyperbolas here described.
Newton's consideration of the semi-circle and hyperbolic cases appears to be influenced by Tartaglia's Nova Scientia (1537), one of the first books applying mathematics to ballistics. Hackborn's On Motion in a Resisting Medium: a Historical Perspective details the Bernoulli affair:
"A “serious error” in Newton's analysis for the semi-circular trajectory in Proposition 10 of the first edition of Principia had unexpected consequences. Johann Bernoulli discovered the error and communicated it to Newton through his nephew, Nikolaus Bernoulli, who was visiting London in September 1712. Newton then tailored a long paste-up correction for the second (1713) edition, already printed, without citing Bernoulli's help... Eventually, responding to a 1717 challenge from Oxford professor John Keill to “[f]ind the curve which a projectile describes” subject to gravity and fluid drag varying as the square of its speed, Johann Bernoulli found an expression for this curve in the case of drag varying as an arbitrary power of speed".
