Actually, he did that in Bag End. The book says that after throwing the Ring into the fire, Gandalf eventually removed it "to the hearth with the tongs, and at once picked it up." Frodo was aghast that the wizard could handle the thing all at once, but it turned out to be cool to the touch, as if a mere wood-fire was not enough even to warm it, much less destroy it.
At the Council, Gandalf does not repeat the experiment, but only there does he actually quote the verses that he saw on the Ring in Bilbo's house.
Actually, the runes were elvish, but the language was the Black Speech of Mordor:
"I cannot read the fiery letters," said Frodo in a quavering voice."No," said Gandalf, "but I can. The letters are Elvish, of an ancient mode, but the language is that of Mordor, which I will not utter here."