Roberts, who was critical of the FA and football's anti-racism group Kick It Out following the John Terry racism case, said there had been a lack of leadership at the governing body.
He said on Twitter: "While Uefa propose a 10 match ban for racism, the FA decides on 5! Who do they consult on these decision? Certainly not the players. The last 18 months regarding equality at the FA has consisted of bad decisions, lack of will, lack of consultation and lack of leadership!"
He added: "Poor decisions on high profile cases, a weak "92" point plan and now 5 games, confirms, it is left to the players concerned to effect change."
The FA's sanction will be the "entry level" punishment for the "least serious" discriminatory offences - including racist and homophobic abuse - with longer bans for higher level offences.
Players found guilty of offences will have to attend educational programmes, and a second offence will be a minimum 10-game ban. Clubs who have two or more players found guilty of discrimination offences within a 12-month period will also face charges and could even have points deducted.
Although there are no openly gay footballers in English professional football, the rules will apply to any player who uses homophobic abuse to any other player.
The new rules come after a review of sanctions following the high-profile cases which saw Chelsea captain Terry banned for four matches and Liverpool forward Luis Suarez for eight games for racist abuse on the pitch.
There is potential embarrassment for the FA however given that it is hosting UEFA's Congress in London next week, and Europe's governing body will submit a resolution recommending that all member associations follow its lead on 10 matches.
FA chairman David Bernstein on Thursday defended the FA's decision not to follow UEFA's lead, saying the five-game ban had been agreed by all parts of the English game including Kick It Out.