''It's time for a change of generations and for some older players to say goodbye to the Serbian jersey,'' said the 29-year-old Vidic, who plays for Manchester United.
Vidic, who returned after an eight-week injury layoff, said it was his choice to take the penalty.
''I'm aware of the consequences of a poorly executed penalty,'' said Vidic, who has scored two goals in 56 appearances for Serbia. ''If I had scored, the game would have taken an entirely different course.''
While Vidic's comments came as a surprise, Stankovic's decision was not unexpected.
''I played many matches, helped the national team and it's time for me to turn to my tasks with Inter Milan,'' the 33-year-old Stankovic said. ''This was my last competitive match for Serbia. I may play another (farewell) friendly.''
Stankovic made his 102nd appearance for Serbia against Slovenia, tying the record set by Savo Milosevic. He scored 15 goals for the national team.
Serbia captain Vladimir Petrovic also appears on his way out.
''I have another two months contract with the football federation,'' Petrovic said. ''In the next few days, we'll have some answers.''