The drawing, published Friday, was the work of award-winning cartoonist Jonathan Zapiro, the BBC reported.
Titled "The Spear to be raised at Social Cohesion Summit," the cartoon features Zuma as a penis looking in a mirror. Next to him is a crude poem.
The "Social Cohesion Summit" is a reference to the Department of Arts and Culture's two-day event in Kliptown, Soweto, this week.
The African National Congress Party said the cartoon is offensive, demeaning and insulting to Zuma, his family and the ANC and demanded an apology from Zapiro and the Mail & Guardian newspaper, which published the cartoon.
"We find it unacceptable and shocking that after the harsh experiences that South Africa, the president and his family has experienced few weeks ago, that Zapiro and the Mail & Guardian will find it appropriate to continue with the insults and hurt to the president, his family and the broader ANC constituency," the ANC said in a statement.
Zapiro and Mail & Guardian editor Nic Dawes have defended the cartoon and its publication, the (Cape Town) Weekend Argus reports.
"Dissident views are essential for real change. Irreverence towards leaders who take themselves too seriously is a vital part of democracy," Zapiro said in a statement Friday.
Dawes said he is willing to defend publishing the cartoon in court.
"Zapiro has a wide degree of latitude to express himself in the Mail & Guardian and it would take a heck of a lot for me to stop him from doing that. I certainly defend his right to engage in the conversation in ways that are sometimes controversial," Dawes said.