Well, I dunno, I think it's not so much proof of his eccentric version of heliocentrism, but the presentation of any version of heliocentrism as truth rather than hypothesis. Cardinal Bellarmine, the Church's presumed expert in the matter, was simply unconvinced by assertions made and evidence presented, when put against the accepted doctrine, as written fairly explicitly in the Scriptures. As you said, Galileo didn't have much solid proof. The Tychonic theory of geocentrism with a mobile sun also explained the phases of Venus, for example. Without having observed stellar parallax, there's not much else you can say - and even if you have a parallax, there are ways to explain around that without having to invoke 'God did it'. In context of a pre-Newtonian era, it's not too difficult to think of alternate explanations.
Galileo had a number of admirers and defenders, up to Urban VIII. The Dialogues, unfortunately, basically attacked everyone else who held different theories, without holding up much else. Most importantly, by using the Pope's own theory (that the universe is in fact geocentric, but made by God to appear heliocentric to human observation) with 'Simplicio' and then striking it down, the Dialogues became a challenge to papal supremacy and had to be put down. It seems Galileo apparently didn't have the social skills to keep his allies and avoid alienating potential supporters.