one minor side note: it does not appear to have been true that most people who considered the matter at all ever considered the earth to be flat.
True or not true, that was the the proof to refute. Many of the same era sought to prove us the center of the universe and for a time logical deduction existed that the sun revolved around the earth. This refute stated most believe we are the center of the universe therefore it must be true. It has long been proved false but there are many even today who believe us the center.
the geocentric model predates christianity by quite a few years - was it Ptolemy or Plato who coined it? can't recall... one o' them musty Hellenes.
anyway, it was the accepted model before the church, and the church found it in accord to a couple of passages in the Scripture which spoke of the sun moving around the earth.
Catholic teachings held it to be true.
On the other hand, those same teachings held the Scripture to be open to interpretation, and of lesser authority as a work (at least in part) of Man than was the reality of the physical universe - which those very same teachings held to be the masterwork of the Lord God Almighty and the supreme test of factuality. Scripture itself could not be in error, but its interpretation could readily be, and such interpretation had to accord with reality, not the reverse.
The church, like any established body, resists change. They required hefty proofs before altering scriptural interpretation. Galileo could not, in 1633, provide those proofs. The fellows he slandered and plagiarized (very good astronomers, and Jesuit priests iirc) could have, but they were rather understandably ticked off at the arrogant old curmudgeon. I don't remember his name, but one thereof iirc was a key figure in bringing charges of heresy (persistend rejection of church teachings) against Galileo, on the grounds that he *knew* Galileo was stretching his arguments beyond what the data could substantiate.
The Pope, Urban somethingorother, was an old friend of Galileo's and quietly his partisan (despite his being lampooned as "Simplicio" in Galileo's book), but...
There was also Spain and France and the ongoing 30 Years War to deal with.
Bad political conditions were Galileo's real bane... along with being a prick and overstating his case beyond what data he had could prove. As I noted earlier in this thread, the inquisition treated him exceptionally mildly.
Anyway, the "proof to refute" or whatever was a traditional church interpretation of the Scripture - believing other than church teachings was heresy... unless you could demonstrate that reality did not beave in the manner that the church taught. If you could do that, as reality trumps scriptural interpretation, then it isn't heresy and the church is duty bound to rethink their interpretation and alter their teachings accordingly.
It's a stupid system, but not so bad as you paint it, and in no way was it Galileo against Stubbornly Ignorant Public Opinion, the way it so often is for scientists against the assorted hounds of bedlam here in the CREVO Wars.
And, again - no educated person from the time of Augustus (at least) thought the earth was flat.