If there were a true demonstration
In 1615, a Carmelite monk from Naples published a tract defending the Copernican system. In his reply, Rome's leading theologian, Cardinal Roberto Bellarmine, congratulated the monk for speaking hypothetically.
Yes, a heliocentric scheme "saved all the appearances" better than Ptolemy's epicycles—the small circular movements he had posited to account for the complicated wanderings of the planets against the background of the stars. But to affirm that "the sun is really fixed in the center of the heavens, and that the earth revolves very swiftly around the sun" was theologically suspect, as it seemed to contradict Scripture.
Bellarmine did admit, "if there were a true demonstration, then it would be necessary to be very careful in explaining Scriptures that seemed contrary," but he had certainly never seen any such demonstration—and did not think that one existed.
What kind of proof would have persuaded Bellarmine that the earth actually, not just hypothetically, revolved around the sun?
Copernicus had offered no physical proof, though his system did provide rational explanations of certain otherwise arbitrary aspects of planetary motions. When in 1610 Galileo with his newly improved telescope discovered the phases of Venus, he demonstrated that Venus had to go around the sun. But while his observation proved Ptolemy wrong, it did not necessarily prove Copernicus right.
As Bellarmine implied, just because assuming the sun is in the center of the universe with Venus going around it neatly explains the observed phases, this does not in fact demonstrate that the sun is actually in the center of the universe. Instead, the earth could be in the center of the universe, with, around it, the orbiting sun, which in turn could carry ...