  Galileo Galilei believed that stars were distant suns whose sizes, measured via his telescope, were a direct indication of distance -- fainter stars (appearing smaller in the telescope) were farther away than brighter ones. Galileo argued in his Dialogue that telescopic observation of a chance alignment of a faint (distant) and bright (closer) star would reveal annual parallax, if such double stars could be found. This would provide support both for Galileo's ideas concerning the nature of stars and for the motion of the Earth. However, Galileo actually made observations of such double stars, well before publication of the Dialogue. We show that the results of these observations, and the likely results of observations of any double star that was a viable subject for Galileo's telescope, would undermine Galileo's ideas, not support them. We argue that such observations would lead either to the more correct conclusion that stars were sun-like bodies of varying sizes which could be physically grouped, or to the less correct conclusion that stars are not sun-like bodies, and even to the idea that the Earth did not move. Lastly, we contrast these conclusions to those reached through applying Galileo's ideas to observations of visible stars as a whole. 
Regarding the Potential Impact of Double Star Observations on   Conceptions of the Universe of Stars in the Early 17th Century