Nope. There are many, many areas of misquotes and misunderstandings about those days. [The one that is most troubling is the claim that Bruno was burned at the stake for his views on astronomy.]
Galileo was active, somewhat, in the Church. He even did some papers such as one on Dante's work. But he wasn't regarded highly as a theologian, but as a great scientist. He was very well liked by the Jesuits, at first. They were located at College Romano (in Rome) and were the more official scientists of their day. Many were brilliant and some were famous mathematicians.
So, when Galileo made telescopes, and gave them to others, he was able to demonstrate a number of things. But the observation that falsified the 2000-year Artistotle model -- formalized by Ptolemy to create practical-use astrological tables, then infused into Catholocism by Thomas Aquinas (he saw the church might look foolish not to adopt such great and suddenly popular reasoning on many things from Aristotle) -- was the telescopic observation of Venus. The Earth-centered model does not allow both a crescent and gibbous phase, but yet it became clear after numerous observations, even with weak lenses and bad seeing conditions, that both those phases exist.
The Jesuits, many who still liked and respected Galileo, agreed that the Aristotle model was falsified -- not that their was any SM back then, but facts are facts after all. Galileo tried to argue that Copernicus was right and that the Sun is the center of the "world" (universe). But their literal religious views, and Council of Trent demands, and other dogma, got them to adopt the Earth-centered Tychonic model where the planets, except Earth, orbited the Sun. This allows Venus to have both phases. So they discounted the wonderful elegance and unification the Copernicus had in explaining things like retrogrades.
Galileo, who had great clout and world acclaim for his discoveries, inventions, and mathematical prowess, got approval to write a book that, he hoped, would turn others within his church to agree with Copernicus (a church canon) and other Church Cardinals who also supported him and Copernicus. The book he wrote argued that tides "proved" the Cop theory and he, unwisely, used the Pope's arguments in a belittling way, though I am sure he did not mean anything personal since the Pope was his friend. The Pope, with tons of more troubling issues, quickly lost patience with Galileo and that's when things started going downhill quickly. The trial ended with Galileo recanting and being stuck at home from then on. [He did write another remarkable book, nevetheless.]
So, if you didn't mind reading all that, the 300 years was the time the Church took to apologize for their mistake regarding their treatment of Galileo. I think it was in 1992. Their adoption, however, of the Tychonic model may have been in only several months. The Cop model was likely consiered more practical to use for planetray calculations so I suspect there was a gradual, but non-vocal, adoption of the Cop model long before 1992.
[Added: There is a term worth learning (teleology) if anyone finds interesting this period -- the birth of science. That time period held that all things in nature came with an intended purpose -- God's purpose. Science as it is today didn't exist as we know it then. Religion, being subjective, was infused with their science, so Galileo, when this was a problem for him, brilliantly argued that religion should be more flexible when discoveries (objective evidence) come along. Interpretations simply needed to be tweaked to make more sense whenever it became obvious a religious or philosophical view was in conflict with the evidence. He was probably the best at trying since he, IMO, was the first powerful person to properly combine today's SM elements including experiments, math, reasoning, etc. ]
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