Fifa president Gianni Infantino has called for the implementation of an automatic forfeit of games for teams whose fans commit racist abuse.
It follows incidents which he described as "totally abhorrent" at Udinese and Sheffield Wednesday on Saturday.
AC Milan's win at Udinese was briefly suspended after alleged racist abuse from home fans towards Milan goalkeeper Mike Maignan.
Coventry's Kasey Palmer said he received similar abuse at Hillsborough.
Infantino said in a post on X that there should be harsher punishments.
"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," he said.
"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!
"The events that took place in Udine and Sheffield on Saturday are totally abhorrent and completely unacceptable. The players affected by Saturday's events have my undivided support.
"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."
AC Milan's players walked off the pitch after France international goalkeeper Maignan reported hearing "monkey noises" coming from a section of the crowd at the Stadio Friuli. The players eventually returned and Milan went on to win 3-2.
"This shouldn't exist in the world of football, but unfortunately for many years this is a recurrence," the 28-year-old told Milan TV.
"With all the cameras present and sanctions for these things, something must be done to change things.
"We all have to react, we must do something because you can't play like this."
Milan and city rivals Inter have both publicly supported Maignan, Serie A said it "condemns all forms of racism", while France striker Kylian Mbappe said "enough is enough".
Former England and Arsenal striker Ian Wright applauded the "solidarity" in the Milan side and urged teams to "keep walking off" when they hear abuse and called for stronger sanctions.
The incident involving Palmer happened during Coventry’s 2-1 victory over next-to-bottom Sheffield Wednesday and led to the match being stopped for several minutes while the match officials spoke to both managers.
Wednesday confirmed the gesture was made by someone in the stands and that anyone "found culpable" would face the "strictest possible sanctions".
But Palmer said he was skeptical that things would change in the game, also writing on X: "Racism is a disgrace... it has no place in the world, let alone football.
"I'm black and proud and I am raising my three kids to be the exact same. I'll be honest, it feels like things will never change, no matter how hard we try.
"[A] couple [of] fans doing monkey chants don't define a fanbase - I appreciate all the love and support I've received."
Anti-discrimination charity Kick It Out has also given its support to both Palmer and Maignan.
"It's up to authorities and clubs to punish those responsible, but if clubs cannot prevent this happening, they too should face consequences," it said in a statement.
"We welcome stronger and more meaningful punishments, as called for by Fifa.
"It cannot be on the players to solve this. We wish it were not the case, but they are already showing courage under extreme distress and emotional trauma. They need support with actions not words."