According to Peter Loyson, who has written a corrective article for the Journal of Chemical Education, Galilio did invent a temperature measuring device called a thermoscope.
In the period 1602–1606, while he was professor at the University of Padua, he made a thermoscope, also called an air thermometer, for measuring relative changes in temperature. He took a “glass bulb with a long slender stem and as narrow as a straw; having well heated the bulb with his hands, he inserted its mouth in a vessel, containing some water, and, withdrawing the heat of his hand from the bulb, instantly the water rose in the neck more than a palm above its level in the vessel. It is thus that he constructed an instrument for measuring the degrees of heat and cold”, according to a letter written by Castelli, one of his pupils. The different degrees of temperature would then be indicated by the expansion and contraction of the air that remained in the bulb, so that the scale would be the reverse of modern thermometers, as the water would stand highest in the coldest weather. About 1611–1612 Galileo substituted spirit of wine for water and later still the Grand Duke Ferdinand II of Florence, a former pupil of Galileo, used colored spirit of wine and reduced the dimension of the tube.
You can get a copy of the original paper here.
Peter Loyson (2012). Galilean Thermometer Not So Galilean J. Chem. Educ, 89 (9) DOI: 10.1021/ed200793g