Many people have already given some adequate answers, especially Evan above me.
Unfortunately, history does not support such a picture. Galileo may not have been guilty of heresy, but he was guilty of several other things: (l) some of his scientific “facts” were wrong; (2) he claimed to have proof when no proof existed; (3) he was unaware of Kepler’s exposition of planetary motion, though Kepler’s book was in his own bookcase; and (4) he had made enemies—bitter enemies—quite needlessly.
All four of those points are true. And that fight he picked with the wrong Pope, turns out Galileo was actually really good friends with Pope Urban VIII… but Galileo was very pig headed and very full of himself. For whatever reason, he had a knack of making enemies, even with some of his best friends.
I haven’t seen this link posted yet, but Catholic sci-fi author Michael Flynn posted a VERY exhaustive, intelligent, well-sourced, and quite funny treatise on the whole subject. Take some time to read it, because it leaves no stone unturned, and you’ll see that Galileo was indeed never “right”. Yes, St. John Paul II apologized for this, because there were things that Galileo said that were right… but he had no scientific evidence to prove his claims. The Catholic Church, specifically St. Robert Bellarmine, wanted to use the scientific method… Galileo did not.
Again, this essay is exhaustive, but if you want to understand why heliocentrism was such a big deal, and why Galileo’s role in all this has been misunderstood since the Enlightenment, look no further than the following link:
The Great Ptolemaic Smackdown by Michael Flynn:
tofspot.blogspot.com/2013/08/the-great-ptolemaic-smackdown.html
Table of Contents (9 sections):
tofspot.blogspot.com/2013/10/the-great-ptolemaic-smackdown-table-of.html