The modern form of this narrative is known as the "conflict thesis" (as in, science and religion inevitably conflict), and it is dated to the work of John William Draper and Andrew Dickson White, who were writing in the late 19th century. Without wanting to discourage more scholarly discussion, the Wikipedia article on this is very good, and covers the main points and works very well, and is exceedingly well cited. For basic information about who they were, what their books were, and what their thesis was, I refer you to it first.

The context of the conflict thesis is important: it is part of the late industrial revolution and it was written by two people who were located in the "burned-over district" of up-state New York, which was the site of the Great Awakening religious movement. Their thesis was an affirmation of the importance of science over religion and tied into things like rejections of Darwinism. It is of a piece of the kind of work that Thomas Huxley was doing in the UK. Huxley, Draper, and White were not just developing this to attack religion, they were developing it as science itself was professionalizing, and they were trying to imbue it with a stronger professional identity. This period is when much of the institutions of what we consider to be "modern science" came into being (and, indeed, the term "scientist" started to be widely used, and many science popularizers/defenders were doing what sociologists of science call "boundary-work," which is to say, rhetorical and sociological "distancing" of themselves from what they were saying they were not, which in this case included religion.

All of which to say is: there is a specifically 19th-century flavor to this reading of history, and Galileo was read from the perspective of defensive 19th century scientists and science advocates, and much of that had nothing to do with the 17th century.