The short answer
No. He was right that the earth moves, but nothing he offered proved it, and the argument he called his proof was wrong. The first proof that a still earth cannot explain came from James Bradley in 1729.
What the telescope could settle
In 1610 there were three ways to arrange the heavens. Ptolemy's put the earth at the centre with everything circling it. Copernicus's put the sun at the centre. Tycho Brahe's kept the earth still, sent the sun round it, and sent the other planets round the sun.
When Galileo watched Venus pass through phases like the moon, he ruled out Ptolemy. But Tycho's arrangement is Copernicus's seen from the earth: the motions relative to us are identical, so every observation fits both. The moons of Jupiter, the phases of Venus, the mountains of the moon, the spots on the sun: all of them fit Tycho as well as Copernicus. That is why the Jesuit astronomers taught Tycho's system.
The tides
Galileo's answer was the sea. The earth's daily spin and its yearly orbit, he argued, add together on one side of the earth and subtract on the other, so every point on the surface speeds up and slows down once a day, and the oceans slosh like water in the bottom of a barge. A still earth, he said, could not produce a tide.
It is wrong at the first step. The earth and its oceans move round the sun together, and the combined motion gives the water no push. It would also give one high tide a day where most seas have two. The real cause is the pull of the moon, as Kepler had said and Galileo had dismissed with contempt.
The real proof
In the 1720s James Bradley found that every star he measured shifts slightly through the year, always towards the direction the earth is moving at the time, the way rain seems to slant towards you when you walk. A still earth cannot produce that. He published in 1729, eighty-seven years after Galileo died.
This site is not connected with the Broadway production of Galileo.