The team suffered three defeats in its three first-round matches: 1-0 against the Netherlands, 6-0 against Argentina and 3-2 against Ivory Coast.
Throughout the World Cup, Serbia-Montenegro had to deal with problems ranging from injuries and squabbles over tactics to confrontations between players at training sessions.
It was also faced with the split of the country, with Montenegro voting for independence in May.
At the World Cup, "we had shown where we stand," Petkovic said, in his veiled criticism of his players. "We showed our true face."
As the successor state, Serbia will keep FIFA membership. Montenegro will apply for a seat with soccer's governing body and hopes to take part in qualifying for the next World Cup.
Serbia-Montenegro's Football Association will formally split in two at a meeting of the soccer body in Belgrade on Wednesday, the AP reports.
Russian MP and former boxer Nikolai Valuev responded to the insults that Ukrainian football fans made against Russian President Vladimir Putin during a Euro 2020 match.