Opening batsman Cameron Bancroft echoed similar sentiments, saying he was determined to earn back the trust of the Australian public.
Cricket Australia last week suspended Smith and David Warner from all international and domestic cricket for a year, while Bancroft was suspended for nine months over the incident during the third Test in South Africa.
All three players have admitted what they did was wrong and a wave of sympathy has been growing for Smith since his public apology, during which he broke down in tears.
The trio had until Thursday to tell Cricket Australia whether they accepted their punishment or would opt for a hearing, as is their right.
Warner has yet to comment but Smith, a golden boy who is compared to Donald Bradman for his batting exploits, took to social media to make clear he would do his time.
"I would give anything to have this behind me and be back representing my country," he tweeted, in his first comments since the emotional press conference on his return to Australia last Thursday.
"But I meant what I said about taking full responsibility as captain of the team.
"I won’t be challenging the sanctions. They've been imposed by CA to send a strong message and I have accepted them."
Smith – the world's number one Test batsman -- was charged with knowledge of the potential ball-altering plan, while Warner was charged with developing the plot and instructing Bancroft to carry it out.
Since the scandal erupted, Smith has lost a lucrative Indian Premier League contract and has been dumped by sponsors.
Bancroft, who used sandpaper to scratch the surface of the ball, said he had informed Cricket Australia he would serve his ban.
"Today I lodged the paperwork with Cricket Australia and will be accepting the sanction handed down," he said on Twitter.
"I would love to put this behind me and will do whatever it takes to earn back the trust of the Australian public."
The Australian Cricketers' Association on Tuesday called for the bans to be reduced, arguing the punishment was disproportionate to previous ball-tampering cases.
ACA president Greg Dyer pointed to the separate International Cricket Council sanction, which suspended Smith for one Test and docked him his match fee.
He also said the contrition expressed by players had been "extraordinary" and should be taken into account, urging a relaxation to allow the men to return to domestic action sooner.
With the World Cup and an Ashes series in 2019, supporters of the players believe they need to be playing state cricket to be in the type of form that could warrant selection.