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.

The paper "Religious Involvement and US Adult Mortality" [Hummer, Rogers, Nam, Ellison] concludes that there is "a seven-year difference in life expectancy at age 20 between those who never attend and those who attend more than once a week."

Now, I'm skeptical of this claim, but it's still a bit shocking.