Shami, whose last test appearance came during India's World Test Championship final loss to Australia in June, had been picked in the squad for the two tests subject to him passing a fitness test.
"Mr Mohd Shami, whose participation in the test series was subject to fitness has not been cleared by the BCCI medical team and the fast bowler is ruled out of the two tests," the BCCI said in a statement.
"Mr Deepak Chahar has informed the BCCI that he will not be available for the upcoming ODI series due to a family medical emergency. The men's selection committee has named Mr Akash Deep as his replacement."
India's tour of South Africa began with a three-match Twenty20 International series, which ended level after the first game was abandoned due to rain.
They will play three ODIs between December 17th-21st, before the first test begins on December 26th in Pretoria, with the second on January 3rd at Newlands in Cape Town.
The BCCI added that batter Shreyas Iyer would only play the first of the three ODIs against South Africa, before joining the test squad.
India's ODI squad: Ruturaj Gaikwad, Sai Sudharsan, Tilak Varma, Rajat Patidar, Rinku Singh, Shreyas Iyer, KL Rahul (captain), Sanju Samson, Axar Patel, Washington Sundar, Kuldeep Yadav, Yuzvendra Chahal, Mukesh Kumar, Avesh Khan, Arshdeep Singh, Akash Deep
