The Brazilian Football Confederation (CBF) named this weekend's penultimate round of matches in the Brazilian championship the "Round against Racism".
"The message the body wants to give... is of protest against the intolerance many players suffer on the pitches of various countries and which ought to be vehemently condemned by the whole football universe," the CBF said in a statement.
"Racism is not resolved with a handshake and none of those who suffer it forget it the next day," CBF president Ricardo Teixeira said in the statement.
"It cannot be justified in the heat of a match nor interpreted as the gesture of a fan. It's something intolerable that doesn't mix with sport," Teixeira added.
"People who don't understand football is for everyone and not just for one race should be banned definitively from sport."
"It's very easy for a person to shake another's hand after a match to say sorry for an insult towards what [the other] holds most sacred and significant. The solution is to find in existing laws a means to give out a very strong punishment," he said.
Blatter's remarks were strongly criticised in Britain where Sports Minister Hugh Robertson and English Professional Footballers' Association chief executive Gordon Taylor called for his resignation.
