Stark accused the Georgian of punching him in the back of the head, causing him to stumble down some stairs and seek refuge in his dressing room after Hertha were beaten to a place in the top flight by Fortuna Dusseldorf.
The German football association - the DFB - ordered a one-year suspension for the 34-year-old, who has the right to appeal. The punishment was announced in a statement on the DFB website.
Kobiashvili's team-mates Christian Lell, Thomas Kraft and Andre Mijatovic have also been banned for their part in the melee which followed the final whistle in the second leg of the play-off fixture in Dusseldorf.
The match on May 15 finished 2-2, with Fortuna winning the tie 4-3 on aggregate.
Lell will miss the first six games of the second division season for violent conduct against an opponent and for insulting the referee, while Kraft will sit out the first five games of the next campaign for insulting the officials.
Mijatovic will miss four games, while Fortuna player Andreas Lambertz will miss his side's first two matches back in the top flight after a 15-year absence, punished for celebrating his side's promotion by holding a flare into the air.
They too have the right to appeal the DFB sanctions and request individual hearings.