Why did Catholic church wait for 350 years to accept "Galileo's idea"?

Question

In 1992 Pope John Paul II officially conceded that the Earth was not stationary - it revolved around the sun

The time that took to be accepted Galileo was right about the kind of relationship between the earth and the sun makes a question about the reason! Was it a political reaction by the Pope to be more popular or a symbolic reaction by the church to try to compensate its dark history or not a real reaction that had not happened before because of church's doubts?

Answer 1

No.

Many volumes have been written about Galileo and the Church, and so an answer on this site cannot do the topic justice. But despite the gleeful reporting by critics of the Catholic Church, the 1992 report was not about astronomy. It was an admission that Galileo had been mistreated and wrongfully convicted.

Soon after his election in 1979, Pope John Paul II appointed a commission to review Galileo's trials. Thirteen years later, as reported in the Los Angeles Times:

The commission found that Galileo's clerical judges acted in good faith but rejected his theories because they were "incapable of dissociating faith from an age-old cosmology"--the biblical vision of the Earth as the center of the universe. "God fixed the Earth upon its foundation, not to be moved forever," says one Bible verse contradicted by Galileo's pioneering notion that the Earth spins daily on its axis and makes an annual journey around the sun.

Unable to comprehend a non-literal reading of Scripture, according to the commission, the judges feared that if Galileo's ideas were taught, they would undermine Catholic tradition at a time when it was under attack by Protestant reformers such as Martin Luther and John Calvin.

Second, a 1984 preliminary report from the commission suggests that Galileo's 1633 trial was prosecuted on a forgery. To simplify, he was accused of violating a 1616 decree that banned him from even discussing the theory. It now appears that the decree entered as evidence was not the same document he had agreed to, and one that was much more restrictive. Galileo was a genius, but an arrogant genius: he had made enemies in Rome.

I would note that heliocentric theories date back to the ancient Greeks, and while Copernicus attracted many critics, both Catholic and Protestant, the Church did not attempt to ban his books for six decades, until they got swept up in anti-Galilean sentiment. Consider also that Galileo's complete works (accompanied by letters stating the Church's criticisms) were published in 1741 with papal imprimatur, and the proscription against them was lifted in 1757, not 1992.

Answer 2

To add to choster's excellent answer, don't forget that Vatican was an intensely political place (then, just as ever). It didn't help Galileo's case that he took the Pope who previously personally supported him:

In the early days of his reign, Galileo had reason to believe Maffeo Barberini's elevation to Pope might lead to a loosening of the Church's opposition to Copernican thought. Pope Urban VIII received Galileo for six long audiences. Although a humanist largely baffled by scientific principles, Urban VIII seemed genuinely interested in Galileo's ideas. U*rban VIII assured Galileo that as long as he remained Pope, the memory of Copernicus had nothing to fear*.
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Pope Urban VIII gave Galileo permission to write a book discussing the contending views of the universe: his Dialogue Concerning the Two Chief World Systems. (src, they have a separate link to sources)

... and basically personally insulted that Pope. In print! (by putting Pope's caricature as an idiot character named Simplicio)

Galileo's biggest mistake seems to have been putting into the mouth of an ignorant, literal-minded character named Simplicio the Pope's own views, offered to Galileo in 1623, concerning God's omnipotence. Urban VIII had argued that an all-powerful God could make the Sun and other heavenly bodies do as he pleased--notwithstanding the laws of physics. In his Dialogue, Galileo provided a response that must have made the Pope feel foolish: "Surely, God could have caused birds to fly with their bones made of solid gold, with their veins full of quicksilver, with their flesh heavier than lead, and with thier wings exceedingly small. He did not, and that ought to show something. It is only in order to shield your ignorance that you put the Lord at every turn to the refuge of a miracle."

Even then, Galileo was strongly supported by one of the 10 judges in the trial, Francesco Barberini - who was none other than Pope's nephew and elevated to Cardinalship by Urban's nepotism.

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Additional factors that forced Urban's hand, as choser's answer alluded to, were due to European religious and political landscape. At the time, Vatican was in the midst of dealing with Protestantism on one hand, and attempts by Spanish Hapsburgs to subjugate vatican to Spain on the other. The latter was critically important - any appearance of deviation from orthodoxy would strengthen the Spanish hand.

As a reminder of the struggle with Spain, recall that Urban was nearly deposed by Spanish Cardinals in 1636