Newton, no less than his frankly materialist or Deist successors, was well aware that the cosmological picture flowing from his own achievement left little room for an interventionist God — an activist, miracle-working being whose constant attention is necessary to the steady functioning of the universe. He sensed that his own brilliant ideas constituted an argument for the deus abscondatus, a conceptual innovation that was soon to become a standard item of skeptical Enlightenment thought. But Newton’s religious traditionalism, unconventional as it was in some respects, found this notion abhorrent because the impersonal God it cautiously endorsed was a far cry from the Biblical Ancient of Days embedded in his own theology. This led him to argue that his own system of the world must be incomplete and that it must indeed be modified to allow a role for an interventionist God whose intermittent action is necessary to keep planets and comets in their orbits. The key point is that this line of thought did not follow from the mathematics of Newton’s mechanics, nor from any sound new physical insight. It was dictated, rather, by the psychological necessity of reconciling his scientific achievement with his pre-existing religious dogma.
We have, of course, no evidence whatsoever that Newton "sensed that his own brilliant ideas constituted an argument for the deus abscondatus," in part because Newton's ideas don't constitute any argument for a deus absconditus, which can't be pulled out of any bit of the mathematics of Newton's mechanics, nor from any of his sound physical insights; and in part because Newton shows no indication of associating any aspect of his system with a notion of deus absconditus or anything like it. The brief discussions of the 'divine sensorium' seem to indicate the very opposite. To the extent that Newton's God could be regarded as distant or hidden, this appears to have more to do with what is usually called Newton's Arianism than his sense of his system. Newton's system of the world was incomplete, which is why we could later have people like Laplace; and Newton, taking his mathematics as accurate, noted that (1) it couldn't explain the first origin of things; and (2) it couldn't account for the long-term continuance of the system. Levitt is right that his conclusion, divine intervention, does not follow from the mathematics or from physical insight; the considerations Newton gives (in the Optics or the letters to Bentley) are teleological. Indeed it is often forgotten that one of the points associated with Newton's famous claim not to feign hypotheses was that hypothesis-feigning was what was done to avoid appealing to teleology (he makes this point explicitly in Optics, Book III, Query 28 -- it also seems to be why the discussion of analysis and synthesis occurs in the context it does later on); in making the claim Newton is, among other things, rejecting the notion that science is merely the study of natural mechanisms. Descartes and Boyle constructed suppositions that allowed them to proceed as if nature were purely mechanistic in character; Newton rejects this approach, and does not assume that an accurate account of the phenomena will be mechanistic. Thus (and it is a 'thus', although the line of reasoning is not simple) Newton affirms design arguments because he thinks design arguments are the sort of thing in which scientific work culminates. The degree to which this depends on pre-existing religious dogma is difficult to determine; it would not have been a widespread religious view at the time, since the rise of the design argument as the most popular argument in natural theology is largely post-Newton (being to some degree spurred by Newton himself), but, then, many of Newton's religious views were not widespread religious views at the time, because there was no real sense in which Newton could be considered guilty of "religious traditionalism," deviating as he does from traditional religious views on many, many points.
There is something a bit odd, given Newton's actual comments and the Leibniz-Clarke debate, in thinking of Newton as the great (if inconsistent) anti-interventionist forebear. I would agree that Newton's interventionism is a bit of intellectual blunder, as Levitt characterizes it; but this is not because Newton was being inconsistent but because Newton had a different view of his scientific work than people came to have later.