Newtonian mechanics was resisted throughout its history, all the way until it was replaced by relativity and quantum mechanics. But the criticism did not so much concern the specifics of his laws of motion, there were few Aristotelians around, and, after all, they were pretty well confirmed by experiments, as their interpretations and "metaphysical" implications.
When Newton published Principia the prevailing view of physics was Cartesian, and the two of them clashed over the course of 19th century. Cartesian physics, was more qualitative, descriptive and "intuitive", in other words, it was more like the natural philosophy of old (e.g. Aristotelian or atomistic). Descartes had the luminiferous aether, and the swirling vortices in it moving things around, in place of Newton's faceless mathematical forces, including the action at a distance for gravity, which even Newton himself disliked. In the early 1700-s Cartesian physics was refurbished by Malebranche and Rohault into an alternative to Newton's. Here is from Iltis's The Decline of Cartesianism in Mechanics:
"Several aspects of Descartes' philosophy of nature were used by Cartesians in the 1720s in attacking mechanical problems. His theory of matter stated that the world was composed of three elements formed from primitive matter. The first element, which formed the material of the sun and stars, consisted of very subtle minute matter capable of moving at enormous speeds and of filling in the small spaces surrounding the other two denser forms of matter. The second element consisted of spherical particles formed from rotation of the original primitive matter. These spheres, the main constituent of celestial matter, moved in large vortices at high speeds and could transmit pressure instantaneously. The third element was coarser and slower and composed the earth and planets.
The properties and action of the spherical aether particles were used by Cartesians in explaining the action of mechanical bodies. As the aether swirled in a centrifugal vortical motion around the earth it caused continual impulses on terrestrial objects. Since the subtle matter moved faster than terrestrial matter and exerted a force toward the center, bodies above the earth would fall toward the earth's center."
By the end of 18th century the Cartesian conception was mostly abandoned, but not the spirit of natural philosophy. The torch was taken up by German romantics and their Naturphilosophie. Göthe, better known as the author of Faust, also penned Theory of Colours, that critiqued Newton's optics and replaced it with a more "poetic" theory, emphasizing the boundary between "light and darkness". Although his theory was rightly rejected by physicists, he made a number of astute observations about the human color perception that are now incorporated into chromatics. Schelling and Hegel rejected mechanistic Newtonian science as such, and planned to replace it with an imagined "better" science, see Why did Hegel hate Newton?
A seemingly technical issue that consumed a lot of attention of many famous people was the vis viva controversy, as to what constitutes the "true" quantity of motion, vis viva (now, kinetic energy) or vis mortia (now, momentum, which Descartes mistakenly believed to be mass times the magnitude of velocity), see What was the vis viva controversy? A more substantive part of it was a dispute between force-based description of physics, that Newton favored, and energy-based one, favored by Leibniz. Leibniz saw both Cartesian and Newtonian physics as too "mechanistic", unable to explain the the "source of the vitality" of matter, which his vis viva, elevated to a metaphysical status, was meant to do.
Newton's absolute space also got its share of detractors. First, Leibniz, who saw space as relational rather than a self-standing metaphysical entity, Berkeley shared similar views and offered an influential criticism of Newton's mechanics, albeit less known now than his criticism of Newton's calculus. Then Kant, who sought to produce a physical construction for the "absolute frame of reference" in almost modern spirit. He associates it with the ideal limit of taking first the frame attached to the Earth, then to the center of mass of the Solar system, etc., see Newton and Kant on Absolute Space by Friedman. These ideas were later be used by Mach to dispense with the absolute space even in the mathematical formalism, and usher in the relativistic conception, adopted and transformed by Einstein.