Excellent that you brought up this fragment of legalese. In an even earlier document dating to Paul V's reign, the Holy Office (i.e., the Inquisition) opined that Galileo, in promoting heliocentrism, was promoting a "formal heresy", but this opinion was rejected by several cardinals. The document you cite, also put out by the Holy Office, reads "suspected by the Holy Office of heresy". The key term here is "suspected". Heliocentrism was never declared heretical infallibly or by an ecumenical council and there's no reason to think it would ever be the subject of such declaration. Furthermore, geocentrism was not a doctrinal position of the Church, though it was held by the astronomers of the day (indeed, heliocentrism was thought to have been refuted by Aristotle who noted the lack of observable stellar paralax that one should expect if Earth orbited around the Sun, something not observed until the 1830s). To clarify the general position of the Church where these subjects are concerned, I offer the following quotes. Augustine of Hippo once wrote "One does not read in the Gospel that the Lord said: I will send you the Paraclete who will teach you about the course of the sun and moon. For He willed to make them Christians, not mathematicians." Cardinal Baronius, who knew Galileo, commented that "The Bible teaches us how to go to heaven, not how the heavens go."

Other documents, such as those where Galileo was supposedly forced to abjure under threat of torture, use language which could be considered legal boilerplate of the day. He was not tortured nor was he in any real danger of being tortured. Galileo, old and frail, was put under house arrest and assigned a servant. As Alfred Whitehead put it, “In a generation which saw the Thirty Years’ War and remembered Alva in the Netherlands, the worst that happened to men of science was that Galileo suffered an honourable detention and a mild reproof, before dying peacefully in his bed.”

Because history is a stage of many actors and many forces, it is also worth noting that Copernicus dedicated "De Revolutionibus" to Paul III, and that his theory was met with interest by Clement VII as well as several cardinals (e.g., Schoenberg). Luther, on the other hand, vehemently rejected heliocentrism. Tycho Brahe argued against it. The point here is that the story of Galileo is repeated ad nauseum by people all too eager to believe in the fairy tale of the big, bad Church and the narrative that the West has been progressively liberating itself from its grip, beginning with the Reformation and followed by the Enlightenment. The more one becomes acquainted with the story of Galileo, the more one sees that it is to a large degree a story of an ornery scientist who was scientifically irresponsible in his aggressive promotion of an unproved hypothesis and who lent fodder to his enemies and made enemies of his friends by engaging in harassment and personal attacks. That does not perhaps justify his condemnation and subsequent house arrest, but all things considered and during an age when brutal punishments were the norm, it seems a little more than ridiculous to hold this up as an example of great conflict between the Superheroes of Science and the Villainous Church. Indeed, it is no accident that science flourished in the West. I refer those interested to Duhem who traces the ideas of physicists such as Newton to scholastic thought.