The term arose from the use of Galileo's trial as an example while making an appeal to pity. It did not come from any particular use by Galileo nor from his preponderance for making appeals to pity.
Of note, Galileo was not actually tortured for his views. By today's standards though, the prosecution of a scientist for advancing knowledge would, in civilized parts of the world, likely be considered an ideological persecution. But even then, there are ethical obstructions to advancing certain kinds of empirical investigation. That said, Galileo was shown the instruments of torture. This was a common practice of escalation after the threat of torture was made, prior to actual infliction of pain.
In April 1633 Galileo was interrogated before the Inquisition. For over two weeks he was imprisoned in an apartment in the Inquisition building. After being shown the instruments of torture, he agreed to plead guilty to a lesser charge in exchange for a more lenient sentence. He declared that the Copernican case was made too strongly in his book Dialogue Concerning the Two Chief World Systems, and offered to refute it in another book.
Just like Lou Gherig didn't know he had "Lou Gherig's Disease" though, I don't think Galileo ever used "Galileo's Argument" even if he made arguments with a logical fallacy of an appeal to pity. For what it is worth, I always understood the "Galileo Argument" as the Galileo Gambit fallacy such that ideas threatening to establishment powers must be valid and sound by virtue of the threat to power. But hey, I learned something new today, so instead of downvoting for clearly demonstrating a lack of research (one more click dude! ;^) ) I say, good question!
If you follow your "Appeal to Pity (the Galileo Argument)" link you will see another reference link which states the following:
Appeal to Pity (the Galileo Argument)
This strategy is an attempt to persuade someone to agree with you or give you what you want by making them pity or feel sorry for you. This fallacy can make the other person seem cruel or insensitive if they do not comply. The connection to Galileo refers to his trials and house arrest by the Inquisition as a result of his scientific views that branded him as a heretic.
On the DonLindsay Archive URL you will find the explanation of the Galileo Argument term coming from the use of Galileo's tribulations as an example in the logical fallacy of appeal to pity:
• Appeal To Pity (Appeal to Sympathy, The Galileo Argument):
"I did not murder my mother and father with an axe ! Please don't find me guilty; I'm suffering enough through being an orphan."
Some authors want you to know they're suffering for their beliefs. For example, "Scientists scoffed at Copernicus and Galileo; they laughed at Edison, Tesla and Marconi; they won't give my ideas a fair hearing either. But time will be the judge. I can wait; I am patient; sooner or later science will be forced to admit that all matter is built, not of atoms, but of tiny capsules of TIME."
There is a strange variant which shows up on Usenet. Somebody refuses to answer questions about their claims, on the grounds that the asker is mean and has hurt their feelings. Or, that the question is personal.