Platini, speaking as he was reelected to a third term leading Europe's governing body, said governments had to stop a return to the 1980s when "hooligans and all manner of fanatics called the shots" in many European stadiums.
The former football star highlighted the 30th anniversary of the 1985 Heysel stadium disaster in Brussels, when 39 people died. Platini played for Juventus against Liverpool in the European Cup final that day.
"Europe is seeing a rise in nationalism and extremism the like of which we have not witnessed for a very long time," Platini told UEFA's annual congress in Vienna.
"This insidious trend can also be observed in our stadiums, as football is a reflection of society. Given its popularity, our sport is a barometer for the ills of our continent. And that barometer is pointing to some worrying developments."
The problem is becoming more acute each year, Platini told a later press conference.
"Its getting worse. There is increasingly more violence, more racist acts, more homophobic acts," he said.
Crowd trouble and racist abuse have particularly hit headlines in recent months.
The Greek government halted the main football league for one week this year because of repeated crowd trouble and pitch invasions. The Greek Cup final between AEK Athens and Olympiakos was called off because of a pitch invasion.
Dynamo Kiev's Europa League game against French side Guingamp was halted for 15 minutes because of crowd trouble. The Ukrainian side had to close part of their stadium for the next round against Everton. They were also fined.
- Feyenoord's Europa League game against Roma in Rotterdam was suspended for more than 10 minutes after a large inflatable banana was thrown on the pitch towards Roma's Ivory Coast winger Gervinho.
"In these battles that we are fighting, we feel as if we have been left to fend for ourselves somewhat. And yet, these are battles that can only be won with the help of the public authorities," Platini added.
The UEFA president said that governments had to take action "so that we can avoid reliving the dark days of a not-so-distant past -- a past where hooligans and all manner of fanatics called the shots in certain European stadiums.
"In recent months, we have all been struck by certain images that I thought were a thing of the past. Some of us experienced that past at first hand. In my case, it was exactly 30 years ago."
Platini was a key player for Juventus in the Heysel stadium where 39 mainly Juventus fans were crushed to death following trouble in the stadium.
Platini called for tougher and European wide bans on known troublemakers in stadiums.
"We need tougher stadium bans at European level and -- I will say it again -- the creation of a European sports police force," he said.