Rising extremism in European society is an "insidious trend (that) can also be observed in our stadiums," Platini said Tuesday at UEFA's annual meeting before being re-elected to a third four-year term as president.
The France great, who witnessed the Heysel Stadium tragedy in 1985 playing for Juventus, said recent violent incidents revived "certain images that I thought were a thing of the past."
"We need tougher stadium bans at European level and, I will say it again, the creation of a European sports police force," Platini said.
In his opening address, the UEFA leader ignored commenting on the upcoming FIFA election and President Sepp Blatter.
During his speech, Platini played down a planned reference to himself as the captain of the UEFA ship, instead describing himself as a teammate and "not the captain of a ship that is being battered by a storm."
The comment harkened back to 2011, when Blatter described himself as the captain of a FIFA ship in troubled waters as the governing body was being rocked by vote-buying scandals.
In a printed version of Platini's speech distributed ahead of time by UEFA, the Frenchman referred to a captain "clinging to the helm for dear life." However, Platini did not deliver the line, speaking minutes after Blatter's own speech was conciliatory toward UEFA and respectfully applauded by delegates.
Blatter appeared at a UEFA meeting for the first time since senior European officials told him last June in Sao Paulo that he had lost credibility while leading the scandal-hit governing body and should not stand for re-election.
"We should not have anger in our hearts," Blatter said Tuesday, calling for unity.
He did not refer to the May 29 election where he faces Michael van Praag of the Netherlands, Luis Figo of Portugal, and UEFA-backed Prince Ali bin al-Hussein of Jordan.
Platini's strongest message raised concern at increasing incidents of violence, and extreme views at club and national team matches.
"Unfortunately, I have two worrying developments to report today," Platini told the leaders of UEFA's 54 member federations.
Platini raised concerns of returning to "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."
"Some of us experienced that past at first hand. In my case, it was exactly 30 years ago," said Platini, who scored the only goal in the 1985 European Cup final in Brussels.
The match was played after 39 fans died in the stadium in rioting provoked by Liverpool fans.
Also, leaders of European clubs, leagues and players' union are set to get greater influence at UEFA, and potentially seats on the executive committee.
"Stakeholders must never be regarded as the enemy," Platini said. "It is time to invite some or all of those branches of the family to join us at the top table."
Potential candidates include European Club Association leader Karl-Heinz Rummenigge, the Bayern Munich chairman, and former West Ham player Bobby Barnes, who leads the European division of worldwide players' union FIFPro.
UEFA members approved the proposal in principle and will discuss it more at a meeting in Malta in September.
UEFA raised its reserves to 558.7 million euros ($613 million), according to its 2013-14 financial report. Revenue was 1.73 billion euros ($1.9 billion) and spending was 1.614 billion ($1.77 billion).
UEFA expects revenue of 4.65 billion euros ($5.1 billion) in 2015-16. Income is rising because of an expanded, 24-team European Championship and lucrative new commercial deals for the Champions League.