Stokes, who is currently under police investigation for his part in a late-night brawl in Bristol last month, tweeted the apology on Wednesday. On the same day, he lost personal sponsor New Balance following his recent arrest over an incident near a Bristol nightclub.
New Balance ended its relationship with Durham all-rounder, saying it "does not condone behaviour by our global athletes that does not match our brand culture and values".
The video emerged following his arrest in September. Price had called Stokes' behaviour "disgusting".
"I foolishly attempted to copy the clip after I saw it a few times," Stokes said on Twitter.
"I should never have done this and I am so sorry," he added.
"It was absolutely not my intention to offend Harvey, Katie or anybody else."
"I have written to both Harvey and Katie to say sorry and prior to this offered to meet in person at the end of last month with a view to also getting behind their petition about online bullying. I do hope I can put this right," he added.
Stokes has been withdrawn from the Ashes squad by the England and Wales Cricket Board (ECB) in the wake of pending investigation into the incident that led to his arrest in Bristol on suspicion of causing actual bodily harm.
The 26-year-old Durham all-rounder was released without charge but remains under police investigation. (ANI)