The 39-year-old East-Enders actor was pictured with what looked like a black eye at the V Festival in Chelmsford, Essex.
He said he is “sweet” but complained his 20-year-old daughter Dani saw the fight.
Dyer tweeted: “No daughter should have to see their old man being toe-punted in the nut by five complete pricks. I’m sweet Bae.”
He also spoke out after Wright, 29, who is married to actress Michelle Keegan, denied he was directly involved in the fight.
Wright wrote in a post on his Twitter feed: “Seen some stories online today about V Festival. I would like to confirm that I wasn’t involved in any physical confrontation.”
Dyer responded: “No. True. Where was ya?? Do you wanna declare who ya snakey 2 Bob mates are?? Especially the 1 squaring up to my Dani.”
Dyer’s fiancee Joanne Mas previously hit out at Wright over the altercation.
She told The Sun: “Mark called him a c*** and was egging on his bunch of idiot friends all day.
“There were five guys against one and Danny still went for them, but the most f****** disgusting thing is it was all in front of my 20-year-old daughter.
“His daughter watched the whole thing, which turned backstage at V into some sort of war zone.”
There has been a long-running feud between Dyer and Wright, with Dyer stating in 2013 that Wright was “everything I hate about this whole celebrity world”.
The spat heated up recently when the actor mocked Wright’s work as a Heart FM DJ and his adverts for Oykos yoghurt.
He wrote: “My worst nightmare would be listening to Heart FM while eating an Oykos yoghurt. It would be like pissing razor blades.”
A source close to Wright said that when the former star of The Only Way Is Essex got to the scene, he shook hands with Dyer to calm the situation down.