Costa was ruled out of the tournament in the US on Friday after scans revealed a torn muscle in his left thigh, Brazil's football confederation said, reports Xinhua.
Kaka, who won the 2007 World Player of the Year award, has rarely featured in Brazil's national team in the past five years.
The 34-year-old currently plays for US Major League Soccer club Orlando City, where he has scored 11 goals in 32 matches since March last year.
Costa's absence is another blow to Brazil, who are already without FC Barcelona star Neymar after the Catalan club said the forward would not be released for both the Copa America and the Olympic Games in August.
Brazil will debut in the Copa America against Ecuador in Pasadena on June 4.