Italy beat Germany 1-0 in a result that meant both teams progressed at the expense of Slovakia in Poland.
The score was the only one that could have sent both teams through but both Italy and Germany deny any collusion.
Fico has written to Uefa about what he called a "farce performed by players from countries advanced in football".
"I believe that you will have the whole issue investigated and Uefa will make rules for the next tournament which will give precedence to a sportsmanlike performance instead of unfair agreements," he added in an open letter to European football's governing body.
"It is disrespectful towards the athletes themselves, fans and also the countries they represent."
Italy's win secured them top spot in Group C, while Germany also finished with six points but bettered the goal difference of Slovakia, who were second in Group A, by a point to progress as the best-placed runners-up.
Slovakia coach Pavel Hapal had also criticised the conduct of Italy and Germany, who rejected any complaints over the issue.
Italy's coach Luigi Di Biagio said any allegations were "ridiculous" and that Hapal should be banned for making them.
Carlo Tavecchio, head of the Italian football federation (FIGC), rejected the suggestion "with indignation".
Horst Hrubesch, the German football federation's sporting director, also denied the allegations.
"I think it's perfectly normal to play cautiously in the last few minutes," he said.
Germany will play England in the semi-final on 27 June, with Italy facing Spain later the same night.
Slovakia's prime minister Robert Fico wants Uefa to investigate his country's group-stage exit from the European Under-21 Championship.