The national boss used the phrase "feed the monkey" during his half-time team talk at Wembley earlier this week, but has since clarified that it was intended as a joke about space rather than a racial slur.
In a letter to FA chairman Greg Dyke, head of the society Peter Herbert said: "To announce that the matter is 'closed' without any action being taken against the England manager is unacceptable and wholly inconsistent with your policies on equality and diversity.
"For Mr Hodgson or anyone else to make an apology if they fail to understand why the words used were offensive or potentially so is not the response of an institution with the resources and profile of the FA.
"The 'innocent remark' made out of ignorance is sadly a common feature of football. We are using the appropriate complaints procedure to urge the FA to provide mandatory 'race appreciation' training and 'cultural capital and cultural intelligence' training to Roy Hodgson and all football managers in the UK."