Is that he was an arrogant ass and often wrong. The Catholic church did not have issue with Galileo's heliocentric view, in fact, the Catholic church has a method to accept and alter their understandings of such natural actions.
The issue is that Galileo's arguments left doubt. Ironically, there were some contemporaries whose work could have aided Galileo's proof of his view. However, he has pretty much dismissed those individuals and their works as wrong. And done so extremely rudely.
The real issue of Galileo's is that he came out postulating "FACT" while by-passing the equivalent of "peer review" for the day. The pope was actually rather fond of Galileo and his work. But refused to acknowledge Galileo's theories as fact, despite his fondness. Then Galileo chose to be a bigger arse. And wrote a book publicly insulting the Pope. It's funny, as we still have this issue in science today over peer review, and early publication statements.
Do you know what the big punishment was? I've read comments deriding the church for executing Galileo. When in truth, Galileo was given a backhanded patronage. He was put on a house arrest. But pretty much had most of his means taken care of, was free to continue his work. It was essentially a public censure.
Ironically, I was unaware of most of these facts until a few years ago. When reading the 1632 series, I started to research Galileo Galilei.
"The matter was investigated by the Roman Inquisition in 1615, and they concluded that it could be supported as only a possibility, not an established fact."
That is not obstruction of science by the church, pope, nada. That is merely saying "Hey, before you declare something as fact, you need to be able to prove it."
Alas, the failure of science here, is to hide this blemish in the failure of history. So we go and teach how Galileo was persecuted for thinking differently. No, Galileo was in trouble for being a rude arrogant ass who couldn't back up his claims.