The hosts lost 3-2 to Egypt in a penalty shootout after their bronze medal playoff ended in a 2-2 draw. The top three teams qualified for the Olympic qualifiers at the World League semifinals in July.
While South Africa could still secure their place in Rio by winning the African Championships in Egypt in October, this option seemed unlikely.
The SA Sports Confederation and Olympic Committee (Sascoc) had previously closed the continental door as part of its “no passengers” approach to the quadrennial Games.
France beat China 5-2 in the final to secure victory with an unbeaten run in Cape Town.