Makgalwa had to been stretchered off the field during Sundowns' 1-1 draw with City in a Premier Soccer League (PSL) match in Tshwane on Tuesday night.
The South Africa under-23 international used his pace to cause havoc for the Citizens defence after being introduced in the second half.
However, Mkhize landed on the youngster’s Achilles tendon and he had to be replaced in an encounter which was played at the Lucas Moripe Stadium.
“This is not sitting well with me. I’ll get hold of him to apologise," Mkhize told Daily Sun.
The tackle left Sundowns coach Pitso Mosimane fuming as Mkhize was only given a yellow card following the incident.
Mkhize insisted he did not injure Makgalwa deliberately with the promising player likely to miss South Africa's upcoming 2019 Caf Under-23 Africa Cup of Nations qualifier.
"There’s no player who goes to the pitch with an intention to injure another player," the experienced defender added.
“I was trying to stop the danger from our team, but unfortunately I injured the young man."
Despite the injury, Makgalwa has been included in the South Africa squad which will take on Zimbabwe in the 2019 U23 Afcon qualifier next month.
The first-leg clash is scheduled to be played on Thursday, 5 September at Milpark Stadium in Johannesburg.