Newton could write in Latin as fluently as he could in English. It is basically what he learned in grammar school, almost full-time (if you include reading the Bible, in Latin and probably Greek, as part of that); he likely learned no mathematics or natural philosophy there, perhaps surprisingly in retrospect, but this would have been fairly common in a 17th-century English grammar school education. Math and science would be learned at university, only after one had mastered the language that would allow you to read them both.

Philosophiae naturalis principia mathematica (1687) was initially an outgrowth of another project, a short (9 page) essay on De motu corporum in gyrum (On the Motion of Bodies in Orbit), which Newton kept reworking and reworking into it ultimately became the first book of Principia (De motu corporum). He drafted and redrafted it several times, and interestingly did not do this entirely alone — for some of it, he wrote in his own hand, but others are written in the hands of his amanuensis, Humphrey Newton (no apparent, or at least no close, relation), to whom Newton dictated portions of it. All of the early drafts we have of both of these are in Latin, including Newton's corrections and amendments to the drafts (of which there were many).

Newton could write freely in both languages, and frequently wrote letters to both (even to the same person). I suspect that if one parsed it out, one could see a difference in when he chose to use Latin versus English, but this is just a hunch based on footnotes here and there. When writing for a learned, international scientific audience, he seemed to use Latin by default. When writing something that was particularly meant to be understood by his English countrymen — such as explaining (which he did at some points) about how he had come up with the ideas in Principia (no doubt part of his relentless priority dispute issues), he wrote in English. But this is just my hunch (it would make sense, anyway).

One interesting and related side-note. Newton's Opticks (1704), published well after Principia, was published first in English, and only translated into Latin later. Why? Newton's main biographer, Richard Westfall, doesn't address this question directly, but his discussion of the circumstances around Opticks is quite telling. Opticks appears to have been directed to an almost entirely English audience: it was presenting what for Newton was very old work (he had done the research and came up with the conclusions as a youth some 30 years earlier), but what was totally unique when it came to contributing to scientific discussions of color and light. Whereas Principia brought about a great many priority questions (e.g., with Hooke over the inverse square law of gravity, and Leibniz over the invention of calculus), Opticks was free of such things (though it was much disputed among the Royal Society mavens when Newton started putting out early versions of it). Westfall essentially says that publishing Opticks was a way of securing Newton's legacy in England, which was core to him being elected as President of the Royal Society. (Opticks came out the year after his election, but was in discussion and development in the years running up to it.) So perhaps that is one of the reasons for writing it in English — it was a different sort of work for a different sort of audience, or patron. One sees similar "politics" of choosing to write in Latin or other languages in early modern Europe as well (notably Galileo, who wrote in either Italian or Latin depending on who he was hoping his writings would influence).

On Newton's education in Latin, and the context of Opticks, and a million other Newton biographical details: see Richard Westfall, Never at Rest: A Biography of Isaac Newton (Cambridge University Press, 1983).

On the specifics of the drafts of De motu and Principia and Humphrey Newton and so on (including images of the drafts), see I. B. Cohen, Introduction to Newton's "Principia" (Harvard University Press, 1971), esp. chapter 3, "Steps towards the 'Principia,'" and chapter 4, "Writing the 'Principia.'"

On the politics of choosing which language to write in at this time period (and elsewhere), see Michael Gordin, Scientific Babel: How Science was Done Before and After Global English (University of Chicago Press, 2015).