I assume the doubts refer to the commentary on Dialogue Concerning the Two Chief World Systems in The Galileo affair: who was on the side of rationality? on Less Wrong. I do not believe that Galileo's case was as weak as the post suggests.
The main objections to heliocentrism and rotating Earth were known since antiquity: the absence of stellar parallax or of fly-off (to which more subtle "Coriolis effect" was added later). There was little that either Copernicus or Galileo could say about the former, other than to speculate that the stars are too distant for the effect to be measurable. The parallax was only measured in 1838 by Bessel. As for the fly-off argument, Galileo's experiments illustrating the relativity of motion, such as descriptions of events on a moving ship, were directly meant to disarm it.
Galileo's arguments were also aimed at undermining geocentric Aristotelian physics generally, e.g. it explanation of falling bodies by their movement to their "natural place", the center of the Earth, mentioned in the post. According to Aristotle, the "degree" of attraction depends on the body's weight, heavier bodies fall faster. This idea was mocked by Philoponus already in antiquity, and Galileo's famous thought experiment with tying two bodies together to show that their weights are moot was also plausible if not logically airtight, see Is Galileo's argument about falling bodies logically flawed?. So, contrary to the psot's author, this was not a particularly strong argument for geocentrism at the time. It is true, however, that Galileo did not have an alternative comprehensive dynamical theory to replace Aristotle's, that would only come later, with Descartes, Huygens and ultimately Newton.
Judging the tidal argument "silly" is anachronistic. Yes, Galileo thought that the tides are due to the rotating Earth rather than Moon's gravity, and that would not work under Newtonian dynamics. But there was no Newtonian dynamics at the time, and the implied effect is real, it explains Earth's bulged shape if not the tides. And the geocentric allusion to "Moon's influence" had no basis in Aristotelian physics of natural places. The connection was based on observed correlations, not on any geocentric insight, and so provides no support either way.
The main contemporary arguments for heliocentrism are not even discussed there. We are not talking about the time a century earlier when Copernicus still had to use epicyclets and excenters to match his system to observations. After Kepler replaced circles with ellipses the match heliocentric system provided was unparalleled. One would need towers of epicycles to get the same precision geocentrically. And trying to incorporate ellipses into something like Tycho Brahe's hybrid system would upend Aristotelian physics anyway. In fairness, Galileo himself mentions neither Tychic system nor elliptic orbits in the Dialogue.
But Galileo's own discoveries of the moons of Jupiter and of the phases of Venus did appear, and suggested that the Earth was no special "natural place". Kepler already speculated about gravity like effects, see Who was first to explain intuitively the inverse square law of gravity?, his speculations were based on Gilbert's theory of magnetism, which Galileo explicitly invokes in the Dialogue.
