Chants could be heard during the 2-2 Group B draw about the killing of Serbians.
The general secretary of the Football Association of Serbia, Jovan Surbatovic, has called for the strongest sanction to be taken.
He told the Serbian state-owned broadcaster RTS: “What happened is scandalous and we will ask [European governing body] Uefa for sanctions, even if it means not continuing the competition.”
Serbia are in Group C and began the tournament with a 1-0 defeat by England on Sunday.
Surbatovic said that he was “sure they will be punished” following Uefa’s decision on Wednesday to cancel the credentials of Kosovar journalist, Arlind Sadiku.
Sadiku made a nationalist double-handed eagle gesture towards Serbia fans during the game against England.
The gesture mimics the eagle on Albania’s national flag, which can inflame tensions between Serbian nationalists and ethnic Albanians, who make up the vast majority of Kosovo’s population.
“We will demand from Uefa to punish the federations of both selections,” Surbatovic added.
“We do not want to participate in that, but if Uefa does not punish them, we will think how will we proceed.”
In a statement the Serbian FA said it had sent a letter to Uefa general secretary Theodore Theodoridis highlighting the “shameful joint chanting”.
BBC Sport has contacted Uefa for further comment.
It is expected that disciplinary proceedings will be brought against Croatia and Albania FAs.
Serbia were fined £12,250 after fans threw objects during the England match.
Serbia and Albania were also fined as fans from both countries displayed banners of nationalist maps, external in their opening matches.
“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, so I appeal to the fans to remain gentlemen.”
Serbia are in Group C and began the tournament with a 1-0 defeat by England on Sunday.
