The player sustained his elbow problem during the recent Twenty20 series against India, which ended in a 2-2 draw.
Bavuma is now facing a two-month break from competitive action which rules him out of the trip to England.
In an Instagram post, the 32-year-old said: "I'm devastated to be missing out on the tour to England and to be out of action for the next couple of months.
"My immediate focus is to ensure that I recover fully so that I can get back out onto the park as soon as possible.
"All the best to Kesh and David who will lead the white ball teams, I know they'll do South Africa proud.
"The boys have my full support for this massive tour."
Spinner Keshav Maharaj will take over as skipper for the three-match ODI series which gets underway on July 19 at the Riverside Ground in Durham.
Batter David Miller, in contrast, will captain the three T20s which will kick off on July 27 in Gloucestershire.
Pace bowler Kagiso Rabada will miss the first half of the tour as he looks to manage his workload. Meanwhile, former Kolpak player Rilee Roussow has been recalled, having last played for his country in 2016.
Gerald Coetzee also features in the squad as he earns his first international call-up as part of the T20 squad.
South Africa's Convenor of Selectors, Victor Mpitsang, said: "The National Selection Panel and I are relishing the opportunity to see what this strong Proteas group will produce in this all-important tour to England.
"With ICC World Test Championship points at stake, the team has everything to play for.
"The T20 format is a high priority for us at the moment because of ICC T20 World Cup coming up in a few short months in Australia.
"We are looking to give opportunities to players that we are interested in seeing and working out the best combinations, while also trying to balance our desire to maintain enough consistency within the set up that the players are used to playing together as a team by the time they reach Australia for the World Cup.
"This series will go a long way in helping us achieve all of those objectives."