Welsh third seed Price, nicknamed ‘The Iceman’, lost his cool in the 6-3 loss.
Price was rattled by an incident after the first set and was criticised on social media when he and Wright did not shake hands at the end of the match.
The 34-year-old initially claimed: “It’s not a rule that you have to shake hands before, after or during a game.”
The former rugby player posted the comment on Twitter at 02:30 GMT on Tuesday after leaving the Alexandra Palace without speaking to the media.
He added: “Banter is acceptable but not when players are fully focused trying to do a job, having just bombed the first set.
“I was upset with what Peter did and think it was out of order considering the circumstances.”
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Price had wound up Wright, 49, before the match by saying he would take the first two sets.
The Scot tapped his opponent on the arm and spoke to him after taking the opener.
When Price levelled at 1-1, he went into Wright’s path during an exuberant celebration.
The Welshman however returned to social media again later on Tuesday morning to say he had let down his family and friends with his reactions.
“I’ve had time to reflect on last night’s game,” he said.
“I’ve been in this position before in games but never in a World Championship semi-final. I overreacted to what Peter did after set one.
“I apologise to everyone who thinks I’ve done wrong and yeah maybe I was a bit OTT (over the top).”
Price also said he had personally apologised to his Scottish opponent.
Wright, who will play Michael van Gerwen in the final on New Year’s Day, said he did not celebrate his victory because he was “focused and concentrated”.
“He wanted me to concentrate and I did just that – and he lost,” Wright added.
Three days earlier, Price had accused Simon Whitlock of “pathetic” slow play after beating the Australian 4-2 to reach the quarter-finals.
Price and Whitlock clashed at last year’s Grand Slam of Darts where the Welsh player was later found to have brought darts into disrepute after also upsetting Gary Anderson in the final, over what the Scot viewed as slow play and over-exuberant celebrations by his opponent.
There was no handshake at the end of that match.
Price was fined a total of £20,000, which was halved on appeal, and after winning that tournament for a second time in November said he wanted to win over his critics.