Not quite. This is what Galileo stated in the Two New Sciences (1638) through his character Sagredo (honest inquirer open to arguments from both sides):
"But I, Simplicio, who have made the test can assure you that a cannon ball weighing one or two hundred pounds, or even more, will not reach the ground by as much as a span ahead of a musket ball weighing only half a pound, provided both are dropped from a height of 200 cubits."
Viviani, in his 1654 biography, reported that Galileo had made the test himself "in front of all the faculty and students assembled", but this is unconfirmed. The common anecdote about the balls dropped from the leaning tower of Pisa grew out of his account. Galileo did study the falling bodies since 1589, and did perform less theatrical experiments. Moreover, already in 1586 Stevin simultaneously released two pieces of lead (1 pound and 10 pounds) from a height of 30 feet, and observed that the sounds of their impacts "could not be separated", see Michon's Final Answers.
Right after the above quoted line, Salviati (Galileo's mouthpiece) gives a famous thought experiment to the same effect. He does stipulate the "same material" though:
"But, even without further experiment, it is possible to prove clearly, by means of a short and conclusive argument, that a heavier body does not move more rapidly than a lighter one, provided both bodies are of the same material, and in short are such as those mentioned by Aristotle.
[...] if a large stone moves with a speed of, say, eight, while a smaller stone moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when tied together make a stone larger than that which before moved with a speed of eight. Hence the heavier body moves with less speed than the lighter; an effect which is contrary to your supposition. Thus you see how, from your assumption that the heavier body moves more rapidly than the light one, I infer that the heavier body moves more slowly... We infer therefore that large and small bodies move with the same speed, provided they are of the same specific gravity."
The specific gravity is proportional to the density. See a discussion of the validity of the argument in Galileo and Prior Philosophy by Atkinson and Peijnenburg. The ship mast experiment, mentioned in the comments, is from the earlier Dialogue Concerning the Two Chief World Systems (1632). It is related, but was meant to argue for the idea of rotating Earth.
Newton stated the equivalence principle more broadly and precisely in 1687 (Principia, Book III, Proposition VI, Theorem VI):
"It has been, now for a long time, observed by others, that all sorts of heavy bodies (allowance being made for the inequality of retardation which they suffer from a small power of resistance in the air) descend to earth from equal heights in equal times; and that equality of times we may distinguish to a great accuracy, by the help of pendulums.
I tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat. I provided two wooden boxes, round and equal: I filled the one with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation of the other. The boxes hanging by equal threads of 11 feet made a couple of pendulums perfectly equal in weight and figure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forward and backward, for a long time, with equal vibrations."
See Cultural History of Gravity and the Equivalence Principle for an outline of further developments.