Prime Minister David Cameron was "appalled" by the fracas at the end of England's Under-21 football match with Serbia, Downing Street says.
He also wants Uefa to impose "tough sanctions" if racism is proved.
England Under-21 defender Danny Rose has called for Serbia to be banned after claiming he was racially abused by fans in Krusevac on Tuesday.
But the Serbian FA has denied any forms of racism towards the England team.
It also said Rose behaved in an "inappropriate, unsportsmanlike and vulgar manner" towards their fans.
FA general secretary Alex Horne has called on Uefa to "take the strongest possible action" against anyone guilty of violence and racist abuse at the end of the match.
"It is also clear that we must defend Danny Rose, who was sent off due to the frustration of being a target of racial abuse.
"No football team should be asked to play in any environment where racial abuse, violence and threatening behaviour is prevalent. We must question the validity of sending a team to Serbia in the future."
And the head of media relations at the FA, Scott Field, said it had evidence which suggested black England players had been targeted.
"We hope Uefa will be dealing with this, as it's a very very serious matter," he said, "we've expressed our deep concern already to them today."
Some fans ran on to the pitch and scuffles broke out as England celebrated a 1-0 win and Euro 2013 qualification.
The government is giving its full backing to a Football Association complaint over alleged racist taunts, a spokesman for Number 10 said, adding that Mr Cameron "was appalled by the scenes that we have seen in Serbia".
"We are determined to stamp out racism internationally and at home and we are giving our full backing to the FA's complaint on this issue.
"Clearly it is for Uefa to investigate this issue but we would expect tough sanctions. If we are going to stamp out racism from football, then it is no good giving derisory fines, as have been handed out in the past.
"It is not good enough to say that people should shake hands and forget about it."
Rose, 22, said he was subjected to monkey chants throughout the game, starting in the warm-up, and then was confronted after England scored in the 90th minute.
"Next thing I know, all the Serbia players were surrounding me, pushing me. I remember getting slapped twice. I got ushered away and that's when I kicked the ball - and then the referee sent me off.
"And then there was more monkey chanting. They should be banned."
Sports minister Hugh Robertson has written to the president of Uefa, Michel Platini, urging him to investigate the incident.
The minister said: "The scenes at the end of the game last night were disgraceful.
"Racism in any form is unacceptable and must be stamped out. We would expect tough sanctions from Uefa on anyone found guilty of racist abuse."
He later told the BBC: "The really important thing is that this is the time to have no more derisory fines, no more punishments that send out the wrong message."
The chairman of the Professional Footballers' Association, Clarke Carlisle, said Serbia should serve a ban from European football.
"Banning them for a start, from any tournament, would be progress but I think if it's significant - if it's a couple of tournaments - then that would cause that nation to address the issue that has deprived them of international competition."
Former England footballer Luther Blissett has also called for Serbia to be banned from football tournaments by Uefa.
"They should get a three, four year ban and that's talking about at club level and international level in my opinion because that it the only way these countries will really start to look at it and say 'we have to do something about it'.
"Because they obviously don't see that they're doing anything wrong by abusing people the way they do."
England defender Nedum Onuoha was racially abused during the 2007 Under-21 European Championship game against Serbia in the Netherlands, for which Serbia were fined £16,000 by Uefa.
A spokesman for Uefa told BBC Sport that it would not comment until it had studied the referee and delegate's reports.