Jones was heard on air saying "the terrorist has got another wicket" when the bearded Amla took a catch to dismiss Sri Lankan batsman Kumar Sangakkara during the fourth day's play on Monday.
An official for TEN Sports, the Dubai-based TV company covering the series, told AFP that Jones had been asked to return home following a complaint from Cricket South Africa (CSA) chief executive Gerald Majola.
"We have taken the matter very seriously and may issue a statement later on Tuesday," the official said on Tuesday. "But I can confirm Deano has gone home."
Jones, 45, told reporters before his early morning departure on Tuesday he regretted making that comment and would send an apology to Amla.
"It was a silly and completely insensitive thing to say and, obviously, it was never supposed to be heard over the air," Jones said.
"I am truly sorry to have caused offence to anybody and the last thing I intended was to be disrespectful."
Jones played 52 Tests and 164 one-day internationals for Australia between 1984 and 1994 and was named the Wisden cricketer of the year in 1990.