Before Ramaphosa spoke, representatives from Cosatu and the SACP delivered messages of support to the ANC and called for Zuma's removal.
In a strongly worded statement the ANC in the Sarah Baartman Region says the sooner Jacob Zuma steps down from the Presidency the better, labelling his efforts to hold on to the position as "unprincipled conduct".
He also addressed the Cape Town water crisis, which has seen residents limited to using 50 litres a day amid fears the city could run out of water in May.
The jibe is in apparent reference to Cyril Ramaphosa's absence from politics to run businesses where he made a fortune before returning to active politics five years ago, becoming Zuma's successor.
"We are determined to rebuild the confidence of our people in the public institutions of our country and to restore the credibility of those who are elected to serve in those institutions", said Ramaphosa, a key negotiator during the transition from apartheid to democracy in the early 1990s who later became a wealthy businessman. Earlier in the week, Ramaphosa is said to have told ANC MPs that immunity from prosecution was not part of their discussions.
Zuma's spokesman did not respond to calls seeking comment.
She posted a happy photo of herself and her husband, seemingly inside a plane, from her Instagram account, firstladytzuma, with the following caption in isiZulu: "My crush, Nxamalala, Msholozi, Maphumephethe, Inkunzi, Emidiwayidwa, Zinyane likaMnyakanya".
He also saluted "such freedom fighters as Sheik Yusuf, Dr Abdullah Abdurahman, Cissy Gool, Alex La Guma, ZR Mahabane, Oscar Mpetha, Philip Kgosana, Ray Alexander, Dora Tamana, John Mtini, Reg September, Dullah Omar, Ben Turok, Annie Silinga, Essa Moosa, Johnny Issel, Jakes Gerwel, Archie Sibeko, Ashley Kriel and many others who were drawn from Cape Town".
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Scott Rolston of Whitestown Police said the department generally would not communicate with ICE for a typical traffic stop. He is also facing a federal case, as he is in the United States illegally, having been deported in 2007 and again in 2009.
Zuma's family said Madiba-Zuma's remarks were "unfortunate".
"Kusazoshipha ukotshi darling. UMsholozi did not join the ANC in 1991, jumped ship nor hip hopped between the struggle and wealth accumulation. He will finish what he started because he does not take orders beyond the Atlantic Ocean #mattersofthestomach (sic)".
"We know you want this matter to be finalised", he told a crowd marking 100 years since the birth of the country's first black president, Nelson Mandela.
Zuma, South Africa's most controversial president since the end of white-minority rule in 1994, has overseen a tumultuous nine years in power marked by economic decline and numerous allegations of corruption.
Zuma had been due to officiate at a diplomatic awards ceremony in Cape Town today, according to his office, but that event has been postponed.
Zuma has expressed his desire to continue ruling the country until the end of his state presidential term in 2019.
The Guptas and Zuma have denied any wrongdoing.