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6 year oldMcGregor threw a metal dolly into the window of a bus carrying fighters at a UFC media day at Brooklyn's Barclay's Center on 5 April.
Chiesa was injured and withdrew from his fight at UFC 223 two days later.
McGregor avoided jail after pleading guilty to disorderly conduct in July.
The claims levelled against McGregor and his entourage are negligence, infliction of emotional distress, assault and battery.
Chiesa, 30, who suffered facial cuts from broken glass, said McGregor's attack caused "serious personal, economic and other injury" and "severe emotional distress, mental trauma and/or bodily harm" in court papers filed to Kings County Supreme Court in Brooklyn, seen by BBC Sport,
The American is also seeking damages from Barclay's Center and its management company BSE Global, accusing them of "failing to protect the plaintiff and other patrons" as employees "failed to adequately perform their security duties."
Chiesa claims he has "experienced pain, suffering and a loss of enjoyment of life" and "economic loss, loss opportunities and economic damages" as he has been "unable to attend to his usual avocation and activities". It adds that he will "continue to suffer in the future."
The amount Chiesa is seeking is not mentioned, but it is more than $25,000 (£19,223) - as he claims it "exceeds the jurisdictional limits of all lower courts which would otherwise have jurisdiction."
Chiesa withdrew from his fight with Anthony Pettis at UFC 223.
He believes he missed out on a UFC title shot at the same event, as the UFC required a last-second replacement for a lightweight title fight that Chiesa was not available for.
Chiesa eventually fought Pettis at UFC 226 in July, but lost in the second round.
Former two-weight champion McGregor, 30, has returned to UFC after almost two years out of the sport and will fight Khabib Nurmagomedov on 6 October.
In summary the claim from Chiesa's lawyers says: "Chiesa has been rendered sick, sore, lame and disabled; that he has experienced pain, suffering and a loss of enjoyment of life and will experience same in the future; that he has been obliged to expend, and will expend in the future, sums of money for medical aid and attention, as well as compensation for economic loss; that he has suffered personal injuries, lost opportunities, and economic damages; that he has been unable to attend to his usual avocation and activities, has suffered a loss and other benefits as a result and believes he will continue to suffer same in the future."
BBC Sport has approached McGregor, the Barclay's Center and BSE Global for comment.