The short answer
There is no evidence he did. No document from his lifetime contains the words. They first appear in print in 1757, more than a century after the trial and a hundred and fifteen years after his death.
Where it comes from
The earliest known source is Giuseppe Baretti, an Italian writer living in London, in The Italian Library, a book about Italian authors that he wrote in English and published in 1757. On page fifty-two he tells the story of Galileo, released after his abjuration, looking up, stamping his foot and saying E pur si move. Baretti gives no source.
Move is not a misprint. It is the older form of muove, the one Dante uses.
What the record from 1633 says
Galileo's collected letters to the end of 1638, the full trial file, and the first biography, written by the devoted pupil who lived with him at the end, have all been searched for the phrase. It does not occur in any of them. That pupil, Vincenzio Viviani, collected every heroic story he could find about his teacher, and this one is not among them.
A painting showing the words on a prison wall is sometimes offered as an earlier witness. It surfaced in 1911; specialists have rejected its attribution to Murillo, and it is now generally dated to the nineteenth century.
Why it survives
Because it is a perfect line. It turns a forced confession into a private victory and makes the trial a story about one man who would not be silenced. The record tells a less tidy story, which is the subject of the book.
This site is not connected with the Broadway production of Galileo.