Your post has a lot of misconceptions in it (the Catholic Church never suppressed the idea of a spherical world — the European world knew that the Earth was round since the Ancient Greeks at least, and the "official" Catholic cosmology has always involved a spherical Earth). But let's just jump to the meat of it: at times, the Church did actively rate certain statements about the natural world as heretical, notably Galileo's advocacy of a moving Earth that rotates around the Sun. The Church astronomers (mainly Jesuits like Christoph Clavius) had lots of reasons for believing that the Ptolemaic model was superior, but modified their belief to the Tychonic model after Galileo's observations of the phases of Venus definitely showed the Ptolemaic model to be erroneous. With the technology and science of the time it was impossible to distinguish between a Copernican and Tychonic models (they are functionally equivalent so long as you cannot observe a stellar parallax to get technical about it). They preferred the Tychonic for philosophical/theological reasons. So when Galileo insisted on preferring and promoting a Copernican model, he was challenging Church political power by asserting a philosophical/theological superiority vis a vis the Church. This came at a time when the Church was dealing with tons of other political challenges to Church authority (the Protestant Reformation). So when he refused to acquiesce — which in his case basically meant, acknowledge that there was no way to tell these two models apart, scientifically — they took it as a direct affront, a heretical statement.

There are more complications involved in the specific case of Galileo, but the above is meant to give you a bit of the gist of the cosmological dispute sitting under a political dispute. The scientific reality at the time was more complex than it tends to be made out to be in retrospect — Galileo was not unambiguously on the side of truth, and indeed many of his own portrayals of the situation are self-serving and not reflective of the Church's actual positions (he continued to portray them as Ptolemaic even after they had changed their model). This does not excuse using heavy-handed, violent techniques to adjudicate their dispute with him, to be sure, though again it needs to be remembered that in the eyes of the church this was essentially a political dispute, a dispute about authority, not about knowledge.

The Church typically has been fine with accommodating discoveries of an unambiguous nature, but it often objected to interpretations that it felt went against scriptural norms. The Copernican model is not a discovery — it is an interpretation of data (a theory). Evolution is in the same boat. The observation of the phases of Venus is more like a discovery in the sense people normally mean — a physical observation that has implications for which theories are plausible and which are not (it rules out Ptolemaic cosmology to a degree that even the Jesuits, once they had replicated it, realized they had to change their cosmological system despite their clear preference for a Ptolemaic one).

Restricteddata has talked a lot about Galileo, I'd like to talk about Evolution.

The Catholic Church has never, officially, denied evolution. When the theory came out... the Vatican made no statement on it. Now, some individual Bishops did criticise it, the Vatican said nothing. The first statement the Vatican officially made about evolution was in the early 20th century, where they said that the theory of evolution is not expressly against the teachings of the Church, but that believing souls have evolved is. As I said, individual Bishops did criticise it, but the Vatican itself has no official teaching on the process of how the world was created (as long as you agree God made it).