Casey had won the first two holes of their duel on Friday and was 2-up until losing the fifth and six holes and then withdrawing.
The Englishman, runner-up in the tournament in both 2009 and 2010, had lost his first match to Thongchai Jaidee of Thailand on Wednesday and fought back late to tie McDowell on Thursday.
World number two Day, the PGA Championship winner, will play American Brandt Snedeker in Saturday morning's round of 16.
(Reporting by Larry Fine in New York; Editing by Andrew Both)