Maybe my first installment was a bit rushed so let me try and improve it a little.
The claim that science originates from Christian thought is vastly overblown.
First because it ignores the scientific contributions of countless non-christian philosophers, scientists, historians and theologians, be they ('pagan') Greek, Muslims, Persians, Indians or Chinese.
Where would modern mathematics be without the Indo-Arabic base-10 numerals? Without the concept of zero?
The second reason is that it ignores the complexity of history, its messiness, and some well-established facts. The comments mentioned Giordano Bruno and Galileo Galilei. The Church did reject heliocentrism for centuries, but also the zero, and later differential calculus and evolution. There's a pattern. In general, the church institution has been scientifically conservative.
The other side of the coin, of this historical pattern -- and that is where it becomes philosophically more interesting, more dialectical than just another sterile fight between theists and atheists -- is that the proponents of heliocentrism and countless other scientific discoveries in the West were also men of the Church, or of some church, until the church rejected them.
In late medieval and early renaissance Europe, when "Western science" was born, a great deal of science and education was done by one church or another. Or more precisely: by certain individuals, certain societies and sub-groups within the vast human groups that one could label "the Catholic Church", "the Lutherans", "the Anglican Church", etc.
It could not be otherwise: nothing is without a cause, and science itself could not come from nowhere. It had to be born in a pre-scientific milieux.
Copernicus obtained a doctorate in canon law and was probably ordained priest, as in 1537 he was one of four candidates for the episcopal seat of Warmia, in Poland.
In Italy, the heartland of Catholicism, Giordano Bruno was a Dominican friar, before he was excommunicated by the Catholics, and later by the Lutherans too... His ideas went way beyond heliocentrism: he thought of stars as distant "suns" around which exoplanets could orbit... That kind of talk was just too wild for the times. Likewise, his theology and hermetology were "out of this world".
Galileo was educated by the Benedictine at Vallombrosa. His Dialogue on the Two World Systems was in fact commissioned by the Pope Urban VIII, a friend from Florence, then newly elected, in an effort to take the latest science into consideration. The Pope wanted some dispassionate, fact-based comparison between 2 hypotheses: geocentrism and heliocentrism. Galileo's Dialogue was that and more: it argued, it mocked geocentrism, and it vehemently promoted heliocentrism. The Pope felt betrayed and unleashed the dogs.
In England, Isaac Newton wrote theological and occult treatises, of the kind that Bruno would have approved of.
The Role of Occultism
The interplay between science, theology and occultism (eg astrology, alchemy and magic) is typical of this historical transition between the essentially qualitative and "enchanted" (tied to religion) medieval world view of scholasticism into the essentially quantitative and "disenchanted" (untied to religion), mathematical world view of modern science.
During the transition, hermetology is a very big deal. Why? Is that simply because of a yet partial, unfinished transition, with people like Newton yet unable to tell magical thinking from science? I would rather take occultism as an indication that people were trying to reconcile Christianity with the cosmological thinking of the ancients, Plato and Plotinus included.
The link is explicit in the case of Marsilio Ficino, one of the first Italian humanists. He was an astrologer, a reviver of Neoplatonism in touch with the major academics of his day, and the first translator of Plato's complete extant works into Latin. (source: Wikipedia)
He was also the first translator of the Corpus Hermeticum, a "collection of 17 Greek writings whose authorship is attributed to the legendary Hermes Trismegistus, a syncretic combination of the Greek god Hermes and the Egyptian god Thoth."
Ficino's translation was a great success that launched a fashion for hermetology -- or hermeticism as we call it today -- in Western Europe.
What Ficino was after though, was not just magic spells and astrological themes, but a form of syncretism between the Bible and the Greeks. He had to try and prove that God had also inspired other cultures, not just Christianity but also the pagan Greeks before it. He had to try and define a religious syncretism between Jesus and Socrates (so to speak), in order for the ancients to be seriously studied, and translated in Europe.
At this point of the argument, the average Christian is usually seized by a holy fear for magic practices, while the average atheist is often puzzled, asking herself:
"Why did Newton, Giordano Bruno, Kepler or Marsilio Ficino even try to save some parts of this mythological nonsense and "wisdom literature" of the ancients? Why occultism? Why didn't Newton found a purely rational (hence atheist) form of science?"
The answer is probably that they couldn't possibly do so, at the time, both politically and psychologically.
It was spiritually impossible, too long a jump.
Nowadays science has made so much progress that it's easy to forget or cancel theology. But by the 16th and 17th centuries science had done very little yet. It was but starting its march, and to begin this journey was to jump in the unknown. It was scary work, spiritually speaking, and one couldn't do it without some faith, without faith that the universe had a meaning, a sense, some inherent Logos, and that us little creatures were able to decipher the Logos with our little-l logos of little creatures.
And so the early humanists needed a theology to found their rationalism. Just a different one than the Catholic dogma. They found it in the ancients, as rehashed by the Italian hermetologists, in Neoplatonism, and then in the Freemasons who stem from the same root.
In summary, Freemasonry and all this "occult stuff" can be seen as a needed historical and spiritual transition between strict Christianity and theism, (and arguably, atheism), a transitional theology that justified a renewed faith in reason, and an interest in ancient (pre-christian) science, philosophy and theology.
Now we are very far from the original claim.