Jovan Surbatovic, General Secretary of the Football Association of Serbia, said a formal complaint had been submitted, claiming that Croatia and Albania fans chanted "kill, kill, kill the Serb" during the 2-2 draw.
He even threatened that Serbia, themselves charged by UEFA for incidents during their defeat by England last Saturday, could withdraw from the tournament.
"First of all, I want to thank our fans for their support in the match against England and I hope we will beat Slovenia," Surbatovic was quoted by Serbia's PTC.
"What happened is scandalous and we will ask UEFA for sanctions, even if it means not continuing the competition. If UEFA doesn't punish them, we will think about how to proceed."
On Monday, the Serbian Football Association was charged by UEFA after their supporters displayed a banner that "transmitted a provocative message unfit for a sports event" and for throwing objects inside the stadium.
That charge came after the Kosovo Football Federation complained to UEFA about "Serbian fans displaying political, chauvinistic, and racist messages against Kosovo" during their 1-0 defeat by England.
"We were punished for isolated cases and our fans behaved much better than the others," Surbatovic said.
"One fan was punished for racist insults and we don't want it to be attributed to others. We Serbs are gentlemen and we have an open heart."
Serbia were due to face Slovenia in Group C on Thursday and a group of Serbia fans chanted "Kosovo is the heart of Serbia" in central Munich's Marienplatz.
Meanwhile, the Football Federation of Macedonia (FFM) also condemned Albania's Mirlind Daku for allegedly directing "nationalist chants" against their country following the draw.
Daku was seen holding a megaphone and encouraging the fans to chant against Macedonians after the game.
"The principles of political neutrality are enshrined in the regulations of international associations such as FIFA and UEFA. The Football Federation of Macedonia will send an official letter to UEFA for this scandalous behaviour," the FFM said.
"We call on Albania footballer Mirlind Daku to publicly apologise for his behaviour, recognising the damage he has caused.
"We also call on the relevant institutions to investigate this incident and take appropriate disciplinary measures to reinforce the standards of behaviour expected of all athletes."
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