Despite his many attempts, Galileo could not prove that the earth went around the sun. However, he was able to prove that the Ptolemeic model was incorrect, after he made telescopic observations of Venus. He discovered that Venus went through a full set of phases, like our moon. This could only happen if Venus went around the sun. In the Ptolemeic model, Venus does not go around the sun, and so would not go through all the phases. The diagram below shows how Venus would only go through crescent phases in the Ptolemeic model. (Recall that the phase of Venus will depend on the relative positions of the earth, Venus and the sun.)
In order to see a "full" Venus, the sun must be between the earth and Venus. A "gibbous" Venus is when the sun is sort of between the earth and Venus. When Venus is at its greatest elongation from the sun, then we would see it as a "half." When Venus was even closer to the earth, we would see it as a crescent. All these conditions happen only when Venus goes around the sun, as in the models by Copernicus and Brahe. In the Ptolemeic model, Venus is always between the earth and the sun, so would only show crescents. Even greatest elongation would not give a half Venus, because the triangle made between the sun, earth and Venus would not be a right triangle.
Galileo's studies of motion also led him to discover the basic idea of inertia. He finally figured out that if an object has a velocity, it will maintain that velocity without the need for any other forces. In fact, it takes a force in order to change a velocity. The reason why objects slow down unless they are pushed is that there is a frictional force that acts whenever two objects rub or slide across each other, and this frictional force is what causes things to slow down. As scientists finally began to accept Galileo's ideas of motion, then the big scientific reasons for rejecting a moving earth disappeared, leaving only the religious.
By this time Kepler had discovered that the planets actually orbitted the sun in an ellipse, but I am not sure why Galileo did not mention Kepler. It should also be noted that Brahe had proposed his own hybrid model of the solar system, with the sun orbiting the earth, and all the other planets orbiting the sun. Brahe's model did correctly explain the phases of Venus, and had the earth at rest in the center. For a while in the seventeenth century, the religious astronomers dumped Ptolemy, but clung to Brahe's as a possible model that kept the earth at rest. Science as a whole, however, had moved on and accepted the heliocentric model. Most scientists accepted the heliocentric theory not because there was proof that the earth moved, but because it was the "prettiest" model that explained the observations.