The History Channel is running a show on the history of astronomy. Wouldn't you think that someone who sets out to do a show on history would have read some of the historical scholarship on the issue? But no. This account of the history of astronomy is essentially the unmodified propaganda of the scientism movement.
First, they failed to distinguish between physical theories and functional theories, treating Ptolemy's astronomy as a successor to Aristotlean physics. This is not correct. Aristlean physics tried to explain how things are. This physics said that the planets occupied perfect invisible spheres. Each planet (including sun and moon) had its own sphere, all with the earth as center. The planets all moved in perfect circles.
Modern scientists laugh at this theory, but as an intellectual exercise the reasoning that led to this theory was no different in principle from modern scientific reasoning. It was an attempt to take an observed regularity and explain it in reasonable terms. Why would the planets move in partly regular, partly random motions? That would not make sense. They must follow some rule. They must actually move in perfect circles, but the relative motions are such that the circles don't look perfect to us. Why circles? Because it was a simple geometric figure. Why did Newton pick straight lines? Why did that seem so reasonable?
Ptolemy didn't have any intention of challenging this physics. Ptolemy's model was just to give accurate predictions of planetary motion. To do this, he added additional motions called epicycles that would not fit in the perfect spheres of Aristotle. Most people who used Ptolemy's model thought that it was physically false. The epicycles of Ptolemy's model were useful for predicting observed behavior, but not for explaining physical truth.
When Copernicus came up with a new model where the planets and earth circled the sun, Copernicus and many others of the time took this heleocentric theory in the same way that they took Ptolemy's theory, as merely a tool to predict planetary motion. There is nothing "ironic" about the fact that Copernicus was a devout Catholic as the show states. It only seems ironic to the show's writers because it doesn't fit their preconceptions. If they were scientific (according to their mythology of what science is) then they would take this observation as a reason to question their preconceptions, but there is nothing scientific about scientism.
Galileo's trouble wasn't that he liked Copernicus's theory, but that he claimed that the heleocentric model was not only a good functional model, but that it was physically true. He claimed that the sun actually was at the center of the universe. Although this wasn't really his problem either. As the show said there was no trouble over Galileo's first tract (the show stipulates that this was "surprising", assuming that their viewers share their own prejudices). In fact, Galileo's first tract was well-accepted by many leaders in the church, including the man who would soon become pope. According to the show, the Church only objected to the second tract because Galileo tried to interpret scripture. There is something to this, but an honest account would also mention that Galileo was abrasive and arrogant and had a habit of making powerful enemies. And that the second tract was personally insulting to the new pope, his former supporter. The Church was certainly not innocent in the Galileo affair, but this was not the one-sided courageous open-minded pure researcher against the evil dogmatic church.
More irritating than the predictable slamming of the Catholic Church, though, was the ridiculous assertion that with his telescope, Galileo "proved" that the heliocentric model was correct. They never explained exactly how the telescope could prove this. The telescope didn't actually give better information about the movements of the planets at all. I think the writers are just confused here. What Galileo did with his telescope is not prove that the heleocentric model is true; what he proved is that the Aristotlean concept of the planets was wrong. Aristotle said not just that the planets moved in perfect spheres, but that the planets were perfect and simple bodies. Galileo was able to make out the craters of the moon and the features of Mars and the moons of Jupiter. This pretty well destroyed that part of Aristotlean physics. So this damaged Aristotlean physics, but in no way proved that the heleocentric physics was correct. Galileo's later work on motion further discredited Aristotlean physics, but it's not fair to say that the Aristotle's theory of planetary motion was actually disproven until Newton came up with a better physical theory.
OK, I'll cut this short except for this mind-boggling statement later in the show: "Without Einstein, we might still be struggling to understand how the universe works." Because, I suppose, we now know exactly how the universe works...