I'm going to echo Eliezer's request for a citation. As far as I know this is simply wrong. First of all, Newton understood that planets interacted with each other gravitationally. Indeed, taking this into account gave slightly better data than the strict Keplerian model. The only planet in this solar system that was predicted based on apparent gravitational influence was Neptune which was predicted based on deviations in the orbit of Uranus. (In fact, people had seen Neptune before but had not realized what it was. Galileo saw it at least once but didn't realize it was a planet (Edit: See remark below)). Uranus wasn't even recognized as a planet until 1781 (some prior intermittent observations of Uranus had marked it possibly as star) and even then wasn't widely accepted as a planet for a few years. Newton died about 50 years prior. So there's no way he could have had any hope of using anomalies in the orbit of Uranus to detect Neptune. The situation gets worse given that the anomalies weren't even recognized until Bouvard's detailed calculations in the early part of the 19th century revealed the discrepancy between the observed and predicted orbit of Uranus.
I thus find Annoyance's comment very hard to understand. It is very hard to attribute Newton's religion as a reason why he didn't use gravitational anomalies to predict the presence of other planets when no such anomalies were at all severe enough in his lifetime to even justify the claim.
I suspect that the commentator may be confusing this with the issue of the stability of the orbits. That is, the orbits of the planets are an inherently chaotic system. Newton had an intuitive but non-rigorous intuition of this problem and suggested that God might step in from time to time to nudge a planet to prevent it from doing something wildly bad.
There are two good books to read on these and related issues. One is Kuhn's "The Copernican Revolution." Despite Kuhn's general philosophy coloring the presentation it gives an excellent summary of the history of astronomy especially around the switch from the Ptolemaic to Keplerian models. The other book to read is Alan Hirschfeld's "Parallax" which focuses on the problem of stellar parallax from ancient times to the modern era and uses that as a general theme to discuss the history of astronomy.
Edit: Just checked. Apparently the claim that Galileo saw Neptune is not completely clear cut at this point. There are two diagrams which show an object in roughly the right places if those observations were taken at the right time but there's no hard evidence. This doesn't impact the overall point substantially.
To paraphrase the Black Belt Bayesian: Behind every exciting, dramatic failure, there is a more important story about a larger and less dramatic failure that made the first failure possible.
If every trace of religion was magically eliminated from the world tomorrow, then—however much improved the lives of many people would be—we would not even have come close to solving the larger failures of sanity that made religion possible in the first place.
We have good cause to spend some of our efforts on trying to eliminate religion directly, because it is a direct problem. But religion also serves the function of an asphyxiated canary in a coal mine—religion is a sign, a symptom, of larger problems that don't go away just because someone loses their religion.
Consider this thought experiment—what could you teach people that is not directly about religion, which is true and useful as a general method of rationality, which would cause them to lose their religions? In fact—imagine that we're going to go and survey all your students five years later, and see how many of them have lost their religions compared to a control group; if you make the slightest move at fighting religion directly, you will invalidate the experiment. You may not make a single mention of religion or any religious belief in your classroom, you may not even hint at it in any obvious way. All your examples must center about real-world cases that have nothing to do with religion.
If you can't fight religion directly, what do you teach that raises the general waterline of sanity to the point that religion goes underwater?
Here are some such topics I've already covered—not avoiding all mention of religion, but it could be done:
But to look at it another way—
Suppose we have a scientist who's still religious, either full-blown scriptural-religion, or in the sense of tossing around vague casual endorsements of "spirituality".
We now know this person is not applying any technical, explicit understanding of...
When you consider it—these are all rather basic matters of study, as such things go. A quick introduction to all of them (well, except naturalistic metaethics) would be... a four-credit undergraduate course with no prerequisites?
But there are Nobel laureates who haven't taken that course! Richard Smalley if you're looking for a cheap shot, or Robert Aumann if you're looking for a scary shot.
And they can't be isolated exceptions. If all of their professional compatriots had taken that course, then Smalley or Aumann would either have been corrected (as their colleagues kindly took them aside and explained the bare fundamentals) or else regarded with too much pity and concern to win a Nobel Prize. Could you—realistically speaking, regardless of fairness—win a Nobel while advocating the existence of Santa Claus?
That's what the dead canary, religion, is telling us: that the general sanity waterline is currently really ridiculously low. Even in the highest halls of science.
If we throw out that dead and rotting canary, then our mine may stink a bit less, but the sanity waterline may not rise much higher.
This is not to criticize the neo-atheist movement. The harm done by religion is clear and present danger, or rather, current and ongoing disaster. Fighting religion's directly harmful effects takes precedence over its use as a canary or experimental indicator. But even if Dawkins, and Dennett, and Harris, and Hitchens should somehow win utterly and absolutely to the last corner of the human sphere, the real work of rationalists will be only just beginning.