"Can't put into words how gutted I am," Brook said on Twitter.
The bout, scheduled for February 23 in Detroit, is Yorkshireman Brook's first shot at a world title.
It is the second time it has been pushed back, with the initial January 19 fight in Los Angeles postponed after Brook injured his ankle.
Brook, 26, became the IBF's mandatory challenger last year.
Alexander is expected to be fit by the end of March, although no date for a rescheduled fight was given.