It was never a common held belief, but there was a lively debate in Europe in the 12th century, where theologians argued the common belief that the Earth was round was superstitious nonsense (they based their argument on the Bible), but the discussion ended with the traditionalist side winning , and the fact that the Earth was round was commonly known among learned men and sailors at Columbus time just as it had been a millenium earlier.
Well, in Dante's divine comedy around 1300 - a theological opus! - it is taken for granted that the reader knows that earth is a ball, and that gravity points to its center.
True, the conviction that only a flat earth is true because consistent with the Bible existed, but it was not a relevant one; not even inside the Church.
The question of the sun orbiting around earth or the other way around was far more important.
But even here, note that Galilei (and others) were not just pursued for stating a disagreeable opinion.
He piqued the vanity of the pope, who was personally interested in his work.
This was his major mistake.
True, several high-positioned clergymen and theologians held the dogmatic position which is today attributed to the Christianity "of the old days" in general.
It seems to me that only Gailei's satirical assault on the pope, and the pope's reply to that have entrenched the Church in its heliocentriphobia.
Otherwise very probably these ideas would have become more and more accepted even inside the Church gradually.
Of course, things could have looked a whole lot different (in both ways) if the conflict would have taken place, say, 50 years before or after ...
The 'Golden Age' Islamic world (if you will, between the start of the Umayyad dynasty and the sacking of Baghdad by the Mongols) did some quintessential steps in the development of the scientific method. Folks like Al-Biruni and Avicanna, anyone?