Bracewell had a scan after experiencing discomfort in his right shoulder in the wake of Australia's innings and 52-run victory in the series opener in Wellington on Monday.
Matt Henry and Neil Wagner are both already with the squad and one of them will deputise for Bracewell as the third seamer in Christchurch, where the second test begins on Saturday.
"Doug has been a consistent performer for us this summer and put a lot of work into his game, so to miss out on the final test against Australia is very disappointing for him and us," coach Mike Hesson said in a news release.
Bracewell took 2-127 in 33 overs in the first test and it was his delivery that should have dismissed Adam Voges for seven before the Australian went on to score 239.
The delivery was ruled a no ball by umpire Richard Illingworth despite television pictures showing Bracewell's foot was clearly behind the line of the crease.
The hosts are already without batsman Ross Taylor (side) and left-arm bowler Mitchell Santner (heel), who both failed to recover from injuries that ruled them out of the first test.
(Reporting by Nick Mulvenney in Sydney, editing by Sudipto Ganguly)
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