The winning question (or, to be accurate, questions) in our Ask the Experts competition has been announced. Congratulations to Richard Ware for soundly testing their knowledge. Just to clarify, the image above is of Galileo, not Richard.
a) How was the first thermometer calibrated?
b) What did Galileo use as a reference? Did he just get his pencil out and start marking the side of the thermometer ""10 ... 20 ... 30 ... 40"" etc?
c) Why did everyone (presumably) accept his findings as pukka? Did anyone query them?
For the answer, see below.
Thinking about these questions, I found I had a less than fresh egg in the larder. I remembered my mother's way of testing the freshness of eggs: fresh ones sink in water; old ones contain gas and float. It seemed to me that perhaps Galileo's mum had taught him the same thing. Being of an enquiring mind, but not an expert cook, perhaps he noticed when told to boil a somewhat doubtful breakfast egg that the egg floated when he put it into the cold water (intending to heat it). When his mum told him you boil an egg by putting it in hot water, he noticed that the egg then sank.
The best comprehensive answer to your questions I've found, based on Middleton (the recognized authority on the histories of the thermometer and the barometer), can be read here. So, I was excited to see there that, around 1603, ""he took a small glass flask, about as large as a small hen's egg"".
In the web article you'll see that Galileo did not calibrate his instrument, merely used it to tell warmer from colder – a ""thermoscope"". Calibration evolved slowly and only became based on fixed points with Daniel Gabriel Fahrenheit (1686-1736), Anders Celsius (1701-1744), and René-Antoine Ferchault de Réaumur (1683-1757).
Pukka-ness comes from continuous review by other reputable scientists and is still the way science advances. So yes, Galileo's mates and all the others developing the ideas at the time would query each other's findings. It seems that the first person to put a scale to Galileo's thermoscope was Santorio Santorio (born 1561). There's a good article, with bibliography, here.
P.S. As an aside, Galileo could, just possibly, have used a pencil. It had been invented by then!