When religion holds the kind of temporal power that allows it to put a teacher under house arrest, that's an argument in favor of the wall of separation of church and state, which of course didn't exist back then.

Most of the more esoteric parts of Christian theology are based on pagan Greek philosophy, including the concept of the Trinity.

John Paul II's commission on this topic itself pointed out that the inquisitors transferred the argument from science to faith. A brief passage from a translation of their report:

Certain theologians, Galileo’s contemporaries, being heirs of a unitary concept of the world universally accepted until the dawn of the seventeenth century, failed to grasp the profound, non-literal meaning of the Scriptures when they described the physical structure of the created universe. This led them unduly to transpose a question of factual observation into the realm of faith.

It is in that historical and cultural framework, far removed from our own times, that Galileo’s judges, unable to dissociate faith from an age-old cosmology, believed quite wrongly that the adoption of the Copernican revolution, in fact not yet definitively proven, was such as to undermine Catholic tradition, and that it was their duty to forbid its being taught. This subjective error of judgment, so clear to us today, led them to a disciplinary measure from which Galileo had much to suffer. These mistakes must be frankly recognized, as you, Holy Father, have requested" (L’Osservatore Romano, November 1, 1992).

And by the way, grammatically, when one translates the passage from the Gospel, one cannot "cast the first stone" because stones are not numbered. One is the first to cast a stone.