Around 1600, the Catholic Church was the primary patron of learning and art by a wide margin. The Pope funded all manner of research. The Church embraced the teachings of St. Thomas Aquinas who, in the 1200s, had shown that God did not create contradictions which meant that science must have supremacy over theology. Science could be proven, but theology must rely on divinely inspired reason.
At the time of Galileo's work on heliocentrism, there were all kinds of wacky theories being bandied about. Human anatomy and medical research was crude and the germ theory of disease was a long way off. Learned men searched for the Philosopher's Stone, a device that would transmute lead into gold. One can imagine the list of crazy theories and inventions the Church was asked to endorse. It's reasonable to assume that the Church was conservative in its support for any particular theory for the same reason that the supporters of the cold fusion study should have been. The price for failure could be a significant loss of prestige.
Galileo's crime against the Church was not one of scientific discovery, but one of impatience. The heliocentric model of the planets was gaining in popularity and the Church, in addition to funding and supporting Galileo, had an independent group of Jesuit astronomers looking into the same thing. The Church asked Galileo not to teach heliocentrism prior to external confirmation and he promised not to do so. In time, his impatience got the better of him and he went back on his word. That was the cause of his trial, not the content of his teaching.
If you look at the problem from the astronomy layman's perspective and you think of all the examples of scientific quackery at the time, it's no wonder the Church, in its role as arbiter of what was true, was slow in recognizing heliocentrism.
|It should have been obvious when the planets were all lined up like this.|