I've been hoping for a while to see a question like this!

The notion of seventeeth-century Ottoman intellectual stagnation has long historiographical roots. As with much of the rest of seventeen-century history, historians of the mid-twentieth century approached the issue within the framework of what at that time was the dominant paradigm of Ottoman history, namely the decline thesis, of which I have written before here. In short, the main questions driving historical research concerned the issue of how the Ottomans fell behind the West, and therefore historians emphasized the ways in which Ottoman science and intellectual life supposedly deviated from the Western norm. Islamic science was recognized as having been extremely sophisticated during the so-called Islamic Golden Age of the medieval period, but was subsequently thought of as having declined for any number of reasons, religious obscurantism and the impact of the Mongol invasions being perhaps the most commonly cited. For the civilizational "rise and fall" or "decline" paradigms of Islamic history, the golden age of the Ottoman Empire in the mid-sixteenth century represented the temporary revival of Islamic civilization, and its subsequent decline required explanation. Contrasting the "innovative" West with the "stagnant" Orient made perfect sense to the historians of the mid-twentieth century, and so the historians making the case for Ottoman decline pointed toward certain pieces of evidence from the late sixteenth and seventeenth centuries to argue that the Ottoman intellectual elite had turned its back on science. Above all, these were 1. the destruction of the Istanbul observatory of Takiyüddin in 1580; 2. the emergence of the puritanical Kadızadeli religious movement; and 3. literary evidence from certain intellectuals complaining about a declining interest in the rational sciences. In a groundbreaking article published in 2008 ("The Myth of 'The Triumph of Fanaticism' in the Seventeenth-Century Ottoman Empire"), Khaled El-Rouayheb attacked all of these foundations of the traditional interpretation of intellectual decline, and called for a closer reading of actual Ottoman texts from this period. His work, much of which has been recently reworked and expanded in his book Islamic Intellectual History in the Seventeenth Century: Scholarly Currents in the Ottoman Empire and the Maghreb has totally revolutionized the field and established a new paradigm for studying the topic. Rather than comparing the Ottomans with Europe, El-Rouayheb has sought to understand Ottoman intellectual life on its own terms. He has found that the seventeenth century actually saw a dramatic increase in interest in rationalist topics. Not a "Scientific Revolution" (and that is another loaded term with its own serious baggage), but a flourishing scientific discourse nevertheless, the contours of which are now beginning to be mapped.

Now, to turn to the question of astrology and the Ottoman elite. While the situation depicted by Orhan Pamuk is clearly based at least in part on the old paradigm of decline, there is certainly an element of truth to it. In the Ottoman Empire as in Europe, astrology was highly popular and Ottoman elites did seek out its practitioners to solicit their services - the Ottoman court even maintained an official position for this purpose, the müneccimbaşı (chief astrologer). The scenario of an astronomer frustrated by his patron's insistence on his performance of astrology is certainly believable. However, astrology never achieved absolute cultural dominance because it was often strongly opposed by the empire's religious officials, who for the most part either frowned upon it or prohibited it outright. It could also be opposed for more scientific reasons, as noted by the poet and madrasa professor Nev'î in one of his late sixteenth-century works:

Question: The questioner asks, "Why should it not be so that the heavenly bodies are the cause of certain events in the mundane world, and why should it not be possible for the intelligent astrologer, by observing the nature of the movements and conjunctions of the stars and the migrations of the constellations, to ascertain the occurrence of certain events before they happen, just as the doctor ascertains the occurrence of a disease ahead of time by the nature of the pulse?" Answer: It is possible that certain occurrences should be the cause of certain other occurrences. Indeed, fire is the cause of conflagration, and this is known by means of the senses. But with regard to this issue, the notion that the planets can be the cause of fortune or misfortune cannot be verified by means of the senses or by rational means.

Empiricism, as it were, was no European monopoly. Astrology was very widespread, but not to the point of snuffing out interest in astronomy, which received a boost in the second half of the century with Katib Çelebi's Cihannüma (which, once it was put into print, became one of the best-sellers of Istanbul's early eighteenth-century book market) and the arrival of the news of new European astronomical discoveries in Ottoman intellectual circles. Unfortunately, much of this has only been very marginally studied, as it was only very recently that this period came out from under the shadow of the paradigm of decline. Now, finally, it's considered a period worth researching. I suspect a really "good" answer to this aspect of your question will take at least another ten years.