The incident happened late in the League Championship match after Holmes was substituted in the 74th minute. The game at Griffin Park in west London ended 3-3.
Derby manager Frank Lampard said: "Duane's upset with the situation. It's not what we want to see in life, not just in football."
In a separate case, Wigan reported a racist message sent to defender Nathan Byrne on Twitter to the police and said: "We find this behaviour abhorrent."
Byrne posted a screenshot on his Twitter account of a message he received following Wigan's 2-2 draw with Bristol City at Ashton Gate.
A statement on the Wigan website read: "Wigan Athletic Football Club are angered by racist abuse received on the personal Twitter account of one of our players, Nathan Byrne, on Saturday afternoon.
"We have reported the incident to the police and will work with the relevant authorities to assist them with their investigations.
"There is no place for discrimination in society and social media is not an exception to those rules. We find this behaviour abhorrent."
Anti-racism group Kick It Out says it will be "liaising with the relevant authorities and offering support to clubs and players involved."
Elsewhere League Two club Northampton Town also reported their players had been racially abused near their hotel in Nottingham ahead of their match with Notts County.
The English Football League (EFL) described the incidents as "reprehensible" and said it would work with Kick It Out to battle racism.
"The EFL was saddened, disappointed and angered to hear of further reports of alleged racism following the conclusion of some matches on Saturday afternoon," a statement from the organisation said.
"It was exceptionally disappointing that this happened as clubs up and down the country were raising awareness of such issues as part of Kick it Out's Week of Action.
"We remain fully committed to continuing to work alongside The FA and Premier League as well as other relevant parties to ensure that football provides a welcoming environment for supporters attending matches and those participating in them.
"We cannot, and will not, accept this type of behaviour inside our stadiums, as we know our clubs won't. We also continue to collaborate with Kick it Out to ensure we eradicate this reprehensible behaviour from our game."
