UEFA has charged Croatia's Football Federation (HNS) for racist chants and symbols displayed by the national team's fans during the Group C match against Italy in Poznan on Thursday.
FARE, an independent anti-discrimination and social inclusion network which works closely with UEFA to identify instances of racism at matches, said hundreds of Croatian fans had racially abused Italy striker Mario Balotelli.
UEFA is also investigating reports that a banana was thrown on to the pitch in the same game, and is looking into racist chanting during two other matches.
"We're very clear what we need; messages from the top, from the leaders of football to say, whatever the environment we're in, whatever the circumstances of where the tournament is being held, that there are some things that are just beyond the pale," FARE executive director Piara Powar told Reuters in a telephone interview.
"We've had that already from some people in UEFA but I think there is nothing like a message that comes from a sanction."
It is not the first time Croatia have landed in hot water with UEFA and world governing body FIFA for racist behaviour by their fans, having twice been fined in the last four years.
The Croatian federation reacted to the latest charge by condemning the chanting and urged UEFA "not to punish the national team".
Fears of racism on the terraces clouded the build-up to the Poland and Ukraine co-hosted tournament, the biggest sporting event to be held in eastern Europe since the end of communism.
Dutch players said they heard monkey chants directed at them during a training session in the southern Polish city of Krakow, two days before the continent's showpiece started.
Czech Republic defender Theodor Gebre Selassie, the first black player to represent the Czech national team, told reporters he had "noticed" racist chants directed at him during his side's game with Russia.
Powar, whose organisation has two "international monitors" present at each Euro 2012 match to identify racist behaviour and far-right banners, said a number of sociological factors were behind the problem.
"A lot of things have come together to make quite a potent mix, some of which has led to violence, some to racism, extreme nationalism, some of it which has just led to hooligans, especially in Poland, wanting to prove that they are still top dogs," he said.
"In many ways I think coming to eastern Europe, bringing a big sporting tournament like this for the first time, in the modern era was always going to throw up challenges.
"We've been (working) in eastern and Central Europe for 10 years or more, we've known the evolution of the story out here. Some of the things that have gone on are a reflection of every day realities in the region."
Russia has been penalised for the displaying of "illicit" far-right banners by their fans in their first two games and Polish and Russian hooligans fought street battles before Tuesday's match in the Polish capital.
"Along with the violence came a lot of indications that the extreme right were influencing what has gone on," Powar said.
"The extreme right influences a lot of youth culture in eastern Europe and a lot of symbolism reflects the extreme right inside stadiums."
Powar, a vocal figure in the fight against racism, said a BBC Panorama programme shown before the tournament highlighting the racist behaviour of Polish and Ukrainian fans, had at least sparked a debate.
"The first point of engagement (after the programme) with the Polish and Ukrainian governments was absolute rebuttal," he said.
"But certainly the Poles, we detected within the space of a week, had changed the public view from denial to one where they accepted they had some issues to deal with and were looking forward to starting their journey to dealing with them."
Powar, who has attended games in Warsaw and Kiev but has not personally witnessed racist behaviour at the tournament, said the "sheer scale of the challenges" in the fight to clamp down on racism meant instances would continue to occur.
"It's always disappointing but at the same time I don't think we will have been able to have done enough work, just because of the sheer scale of the challenges to prevent to what has happened," he said.
"If at the end of the tournament we feel there are issues that have been left unsanctioned then we'll come back and have a conversation with UEFA and we will be clear in our public statements that more could have been done."