The head of the world governing body said there was no place for discrimination of any kind in football or broader society.
"The events that took place in Udine and Sheffield on Saturday are totally abhorrent and completely unacceptable," he said in a statement.
"The players affected by Saturday's events have my undivided support."
Fans aimed monkey chants at AC Milan and France goalkeeper Mike Maignan during Milan's dramatic 3-2 win at Udinese, with the game temporarily halted.
Support from Mbappe
In a strongly worded statement on X, formerly Twitter, the France team said: "You have all our support Mike Maignan. The FFF (French federation) totally condemns all racist acts."
France captain Kylian Mbappe also offered his backing to his international teammate.
"You are very far from being alone Mike Maignan, we are all with you," wrote the Paris Saint-Germain star.
"Still the same problems and still NO solution. Enough is enough!!!!! NO TO RACISM" he posted on X.
Coventry midfielder Kasey Palmer accused Sheffield Wednesday fans of doing the same towards him during their English Championship clash, which his team won 2-1.
"We need all the relevant stakeholders to take action, starting with education in schools so that future generations understand that this is not part of football or society," Infantino said.
"As well as the three-step process (match stopped, match re-stopped, match abandoned), we have to implement an automatic forfeit for the team whose fans have committed racism and caused the match to be abandoned as well as worldwide stadium bans and criminal charges for racists.
"FIFA and football shows full solidarity to victims of racism and any form of discrimination. Once and for all: No to racism! No to any form of discrimination!"
At Udinese, referee Fabio Maresca stopped play during the first half and a livid Maignan stormed down the tunnel with his teammates.
"They're ignorant people... You can be booed or whistled when you're away from home, that's normal, but what happened today has no place in football," Maignan told Sky Sports.
Play resumed after about five minutes.
Italy, a country governed by a coalition led by the far-right Brothers of Italy party, is rife with fascist football fan groups, in particular among the hardcore "ultras" who make most of the atmosphere at stadiums.
Last week Lazio were hit with a one-match stand closure after supporters directed monkey chants at Romelu Lukaku during their team's Italian Cup win over local rivals Roma.
In Sheffield, Palmer labelled the abuse "abhorrent and wholly unacceptable", with the match paused as the referee spoke to both managers on the touchline.
In a statement, Sheffield Wednesday said they were "shocked and saddened by the racist gesture from the stands reported by Sky Blues player Kasey Palmer."
"Both clubs roundly condemn any form of discrimination and abuse, and underline that there is no place for this kind of behaviour in football or our wider society," it added.
"We will work together with the relevant authorities and anyone proven to be culpable will face the strictest possible sanctions from both Sheffield Wednesday and the law."
