The question of when science became science is a difficult one. Thomas Kuhn considered science a process of puzzle-solving, where the puzzles relate to fitting a worldview, or "paradigm," with observations of nature. His book The Structure of Scientific Revolutions has been pretty influential among historians of science (not as much with scientists themselves, however). I recommend reading it to get a very useful perspective on what science is and how it functions. He offers several examples from the field of chemistry, so it might serve you there, also.

As far as chemistry is concerned, you've picked a decent place to draw the line. There's never an abrupt change from non-modern to modern science (or anything). Keep in mind also that terms like "modern" are very vague and don't do much useful work for you. What characterizes modern chemistry as opposed to "pre-modern" chemistry? If you can't answer that question neatly then I can't answer your question neatly.

Further, Newton's contributions to chemistry are usually classed pretty firmly in the "alchemical" camp. It was physics he was famous for revolutionizing, not chemistry. For chemistry, you might look a little bit earlier to Robert Boyle, still famous for Boyle's law. He did a lot of experiments with air pressure (or "spring of the air" as he called it), including his famous air-pump.

The work of Newton and Boyle was based on a pretty recent consensus in Europe about the corpuscular nature of matter. Aristotelian understandings of matter, dominant in the later middle ages, took stuff to be made up of continuous elements, each of which had natural inclinations (fire to rise, earth to fall, for instance). The corpuscular view of matter instead held that things are made up of tiny, inert corpuscles (basically atoms). These corpuscles don't have natural inclinations in any particular direction, and matter can therefore be explained without reference to Aristotle's elemental theory. Galileo was a proponent of the new corpuscular theory (60 or so years before Newton). You can read about a really interesting dispute he had with an Aristotelian philosopher about buoyancy in Galileo Courtier. The nature of matter was one of the central questions of the dispute.

This change in scientific understanding of matter was part of the "scientific revolution," and both chemistry and physics changed a great deal between 1600 and 1750. So no, I would not consider chemistry younger or newer than physics.

One thing to be aware of is that philosophy used to refer to a lot more than it does today. During the early modern period most of what we would call science today was lumped under the broad heading of natural philosophy. Newton's book was titled "The Mathematical Principles of Natural Philosophy", what we consider a contribution to physics he considered a work of natural philosophy. Almost all intellectual inquiry came under one of two headings before the 17th century, Natural History and Natural Philosophy. During this period people became aware of the success of what they called experimental (natural) philosophy, which was largely the emerging scientific method, but it would take some time for it to be considered an independent discipline. As Sebatinsky noted Robert Boyle is one of the founders of modern chemistry, and he was working in the same milieu as Netwon in and around the early Royal Society.