Brazilian striker Neymar accused Scottish fans of racist abuse after being repeatedly jeered during the match at Arsenal's Emirates Stadium.
A banana was also thrown onto the pitch although police later said the item of fruit was thrown by a German tourist who had "no racist intent."
Neymar's allegations outraged Scottish supporters and authorities, who cited the Tartan Army's positive reputation, and now SFA chief Stewart Regan has written to the Brazilian federation to seek an apology.
"It is very disappointing that this episode threatened to overshadow an entertaining match, played amidst a real carnival atmosphere created by supporters of both teams," Regan said.
"Scotland and Brazil have played many times over the years, each time in a colourful and friendly environment, and we feel that it's necessary to draw a line under Sunday's events.
"Scotland's supporters were genuinely upset and annoyed that the unfounded allegations levelled against them threatened to sully their reputation as fair, good-humoured and respectful spectators.
"As the matter has been resolved, and Scotland supporters cleared of any involvement, we felt it was only right that we acted on behalf of our supporters to request an apology either from the player involved, or the Brazilian FA."