Galileo's dispute with the Catholic Church about his scientific discoveries is famous, but the truth is much more complex than is generally understood (following material largely from "The Church's Attempts to Dispel the Galileo Myth" by G. V. Coyne and "The Sleepwalkers" by A. Koestler).

From Chris Madden http://www.goma.demon.co.uk/moon/galileo.html

The dispute occurred against a background far removed from our present context. Science had for centuries been philosophically detached from observation - the term "to save the appearances" was widely used at the time to describe science that conveniently simplified observation but had no deeper significance. Galileo was far in advance of his contemporaries in a modern view of attempting to explain true physical reality rather than just providing a convenient structure to classify observations. The Church was in the midst of contending with the Reformation, a position that tended to push it to conservative positions. Many of the scientists of the day were Jesuit priests, so the Church represented an independent scientific force as well as its role defending religious doctrine. Of course, the Jesuit scientists approached science from the standpoint of "saving the appearances," so Galileo's effort to find true physical reality was philosophically totally incompatible with their views. Although there were many reasons for the dispute, this irreconcilable difference in how science was approached lay beneath everything.

The shortcomings of the Copernican system were well understood. Kepler, for example, had written to Galileo in 1597 imploring him to look for parallax as a way of establishing that the earth orbits the sun, concluding with "Even if we could detect no displacement at all, we would nevertheless share the laurels of having investigated a most noble problem which nobody has attacked before us." Galileo also tried to hide the complex epicycles required by the Copernican system -- more about that later. Thus, there was at the time no absolute proof of the correctness of the Copernican view, and considerable evidence against it. Nonetheless, Galileo's discovery of the moons orbiting Jupiter and the phases of Venus gave the kind of support to the Copernican system that would cause many a modern scientist (and Galileo was in many ways the first such) to leap to the conclusion that it pointed toward the true physical nature of the solar system.

The Church reassured Galileo repeatedly that he was free to explore the Copernican system as a hypothesis that "saved the appearances" - that is, fit the motions of the planets - so long as he did not promote it as absolute proven truth. This position was defensible, given the lack of measurements of parallax for example. However, it also represented the philosophical divide between medieval and modern science. Galileo's approach made reasoned discussion of philosophy impossible. His early works had a polemical style, laced with satire and virulent attacks on rivals and occasionally a scrap of brilliance, as we will illustrate below. His antagonistic style contributed significantly to the difficulties. .

A leading figure on the side of the Church was Cardinal Bellarmine, who conveyed to Galileo through an intermediary in 1615: "..as to Copernicus, there is no question of his book being prohibited; the worst that might happen.. would be the addition of some material in the margins..to the effect that Copernicus had introduced his theory in order to save the appearances, or some such thing..and with a similar precaution you may at any time deal with these matters." Galileo strongly rejected this proposed compromise, insisting that the Copernican system was absolutely true; at the same time, he  arrogantly elected to refute the Cardinal's interpretation of the Bible regarding the motion or lack thereof for the sun. His excuse for not producing proofs for his view was that the others (presumably including the Cardinal) would not be sufficiently intelligent to understand them: "To me, the surest and swiftest way to prove that the position of Copernicus is not contrary to Scripture would be to give a host of proofs that it is true and that the contrary cannot be maintained....But how can I do this, and not be merely wasting my time, when those peripatetics who must be convinced show themselves incapable of following even the simplest and easiest of arguments."  This early part of the dispute led to the decree of the Holy Office of 5 March, 1616, which did not address Galileo directly but ordered that certain passages in books by others be altered so that they did not claim that the Copernican system was consistent with the Bible - the Holy Office wanted that determination left to itself. The application of the decree appears to have been intentionally limited, but its restrictions were clearly aimed at Galileo's approach even if he was not named.

Developments following this decree were strongly influenced by Galileo's style. A Jesuit father, Horatio Grassi, advocated the correct view that comets move in regular orbits like planets. Galileo's own copy of this treatise has survived, and its margins are profusely annotated with terms like "piece of asininity," "evil poltroon," and "ungrateful villain." He attacked Grassi publicly in a manuscript that he persuaded a former pupil to sign, although it was in Galileo's handwriting. At one point he wrote: "You cannot help it, Signor Sarsi, that it was granted to me alone to discover all the new phenomena in the sky and nothing to anybody else. This is the truth which neither malice nor envy can suppress." His sarcastic attacks alienated Grassi and many other powerful Jesuit scholars. He continued his work, and continued also the controversy over whether the Copernican system was to "save the appearances" or represented physical fact. Eventually Pope Urban suggested a solution to the difficulty: even if the Copernican system explains the motions of the planets, it may not be true because God is all-powerful and might have produced the same appearances by some entirely different means that are not understood.

Galileo finally published his work in "Dialog on the Great World Systems," in 1630 and at the age of 65. It was written as a discussion among his brilliant mouthpiece, Salviati; an intelligent but uninformed straight man, Sagredo; and a good-humored fool, Simplicio. Galileo managed to maneuver around the review by the Church and had it published without approval. When it appeared, the Pope's arguments on how an all-powerful God might have produced the planetary motions in some other way emerged from the mouth of Simplicio! At the same time, the arguments of Copernicus had been made superficially more attractive by claiming that the planets moved around the sun on simple circles, leaving out any pretense of actually fitting the observed motions. The system was defended with a number of arguments that were obviously invalid, as well as with the ones now widely recognized as being definitive.

The combination of circumstances, including that the book was in violation of the decree of 1616, led quickly to the famous trial of Galileo. Amazingly, despite the clear printed word, he tried to lie to the Inquisition, claiming that he had written to refute the Copernican system! Eventually, he was interviewed privately and firmly and forced to acknowledge that this claim was false. This admission caused the sudden loss of confidence that led to his final appearance on the stage of history as a broken old man. Painting by Cristiano Banti (from http://web.hao.ucar.edu/public/education/sp/images/trial.html)