Not on astronomical grounds, but on philosophical ones. His observations (mountains on the moon, phases of Venus, sun spots, moons of Jupiter) led Galileo to the conviction that the celestial bodies were made from the same materials as terrestial ones. Remember, aether wasn't just the stuff that filled the space between the celestial bodies, it's also what those bodies were supposedly made of, allowing them the rapid daily motion you have to posit if earth is not supposed to rotate. One of the arguments formerly brought forth against terrestial motion gets thereby neatly turned on its head.
The problem was that while personally I find this line of reasoning rather convincing, it's not a smoking gun, which led Galileo on his quest for more direct observational evidence, eg his (incorrect) theory of the tides.
The Copernican model held that planets moved in circular orbits at uniform angular velocity--which is sufficiently wrong that it required only a few less epicycles than Ptolemaic astronomy. You can't fall back on "it's more accurate" since the results weren't really any more accurate. Yet, for this inaccurate results, a Copernican would have to explain:
* The lack of stellar parallax
* How the Earth can rotate around the sun
* Why we don't fall off the planet
* Why the rotation of the planet doesn't throw us off the planet
The Copernicans had no good answers for these questions--the latter were eventually explained by Newton's Law of Gravitation which proved to cement the transition away from geocentrism (the accuracy of Kepler's elliptical orbit also provided another major impetus for the transition, since it didn't need any epicycles. Incidentally, Galileo saw Kepler as a Copernican heretic and viewed his model of planetary orbits as utter rubbish).
Stellar parallax was resolved when we actually found it--in the 1700s. The Copernican attempts to explain it were weak and sometimes involved actual heresy on the nature of God, declared as such by the Catholic church 100s of years prior. Tycho Brahe gave a fairly comprehensive, scientific denunciation of Copernican astronomy based on his stellar observations.
Incidentally, your number for the deaths caused by the Catholic Church during the Inquisition is off by over an order of magnitude. The Spanish Inquisition appears to have executed only around 2000 people, and that for all of the major executions put together doesn't seem more than around 10,000 people. In general, the Inquisition's goal was mainly getting heretics to convert, not executing them: only around 5% of guilty sentences produced executions in the first place.