Captain van Niekerk, who has made 194 appearances in all formats for her country, was unable to complete a two-kilometre run in the minimum required time.
The 29-year-old wrote on social media on Tuesday that she was "broken" by her omission.
Head coach Hilton Moreeng revealed that Kapp is still committed to the squad and the leave was to help her prepare for the upcoming World Cup.
"We granted her leave so she can come to the World Cup and be ready to play," he told the press.
Kapp, 33, is set to miss the final T20 of a triangular series against India before rejoining the squad ahead of the World Cup on home soil.
"Emotional couple of days, I am a wife FIRST and then a cricketer," Kapp posted on Twitter.
"Thank you CSA for affording me the opportunity to be home to support my wife and see the family, as I mentally refresh and prepare for the T20 World Cup. It has always been my biggest honour to represent my country."