The short answer
Probably not as told. The story of two weights released from the tower before the whole university first appears in a biography written in 1654, sixty-five years after the events, by a pupil who met Galileo only at the end of his life.
Where the story comes from
Vincenzio Viviani came to live with Galileo in 1639, aged seventeen, and stayed until his death. He loved him, and in 1654 he wrote the first life. The leaning tower is in it. So is the lamp swinging in Pisa cathedral that supposedly gave Galileo the pendulum. Neither story appears in anything written before it.
The lamp that now hangs in the cathedral is usually dated 1587, after Galileo had left the university, though some lamp hung there before it.
Viviani was writing a saint's life. He wanted the beginning to match the end.
What Galileo did argue
The argument itself is real, and he kept it for forty-eight years. In his last book, Two New Sciences (1638), he showed that a falling body covers distances in successive equal times in the ratio one, three, five, seven, so that the total goes as the square of the time. He could not time a free fall with the clocks he had, so he slowed it down: a ball rolling down a gentle slope is falling, diluted, and the same ratios appear at every angle.
This site is not connected with the Broadway production of Galileo.