According to the website of the New York City Department of Parks & Recreation, the statue is a bronze replica of an iron and copper piece by the German sculptor Karl Fischer (1802-1865) from around 1832. (The website also says that Goethe Society of America acquired the statue in 1876 and that it was displayed at the Metropolitan Museum of Art, before being moved to Bryant Park in 1832.)
I have been unable to find Karl Fischer on the Web and am beginning to wonder if the statue is not instead a copy or a re-making of the Goethe bust by Christian Daniel Rauch from 1820, now in the Goethe Museum in Düsseldorf. Obviously some art historical detective work is in order to straighten out what seems like a misattribution.
While I am pleased to see Goethe in Bryant Park, the statue is horribly positioned at the moment, as can be see from the above photo, taken by me this past summer. What an ignoble position for Goethe, facing the merry-go-round and its circus-y sound. If anyone would like to join me in requesting that the statue be turned to face the green in Bryant Park, please let me know.
Rauch photo credit: Bradsculptor's Blog