On a chaotic morning for Australia, Smith was ruled out and Josh Inglis drafted in to replace him as one of four changes in their team for Newlands.
Scott Boland, Nic Maddinson and Beau Webster have all been dropped, with Nathan Lyon, Cooper Connolly and the fit-again Cameron Green all coming in.
Marnus Labuschagne has held his spot, in what may be his last chance to stay in Australia's Test team ahead of the home summer, and Matt Renshaw will open with Travis Head.
The four changes are the most Australia has made mid-series since the ball-tampering scandal on their last visit to Cape Town in 2018.
Smith had been due to play his first Test at Newlands since that match this week, before being struck down by his neck issue.
He did not join teammates on the team bus from the hotel on Tuesday morning, instead arriving ahead of time and going to the Newlands nets behind the ground.
The 37-year-old wore a heat pack on his neck and appeared frustrated, before speaking with coach Andrew McDonald and selectors' chairman George Bailey.
He then remained in the Australian changerooms until some 30 minutes before the toss, and did not participate in slips catching when he did come onto the field.
"Unfortunately Steve's neck is not quite right," captain Pat Cummins said.
"He had another little try this morning, but wasn't happy with how his neck was feeling, so we have had to look elsewhere."
Smith has battled neck stiffness since his first innings in Gqeberha last week, which impacted his batting in Australia's thrashing at the hands of the Proteas.
He only batted against throwdowns for about 20 minutes during Australia's main session on Sunday, before a 45-minute hit on Monday.
The vice-captain also looked frustrated through the later parts of that session in Cape Town, with the series locked at 1-1 headed into the series decider.
Cummins also said that at the toss that the decision to leave Boland out was "very hard", after he was arguably Australia's best in the second Test with a five-wicket haul and lengthy innings as a nightwatchman.
South Africa have made one change, with Gerald Coetzee coming in for the injured Anrich Nortje.
AUSTRALIA: Travis Head, Matt Renshaw, Marnus Labuschagne, Cameron Green, Josh Inglis, Alex Carey, Cooper Connolly, Mitchell Starc, Pat Cummins (c), Nathan Lyon, Josh Hazlewood.
SOUTH AFRICA: Aiden Markram, Marques Ackerman, David Bedingham, Temba Bavuma (c), Matthew Breetzke, Ryan Rickelton, Wiaan Mulder, Â Marco Jansen, Simon Harmer, Gerald Coetzee, Kagiso Rabada.
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