The video was produced by one of Pique's companies, leading to criticism of the Spain international.
However, the 31-year-old said he had no involvement in the production, other than suggesting the idea to Griezmann.
"It's one of my companies, the idea came from months ago when I spoke to Griezmann," Pique said after Spain's 3-3 draw against Portugal at the World Cup on Friday.
"He told me there was a chance to come to Barcelona and I told him we'd record it, that it could be great content for people to see someone take such an important decision.
"From there, I wasn't involved, obviously. I wasn't filming to see what he had to say and people saw that until the last day he wasn't sure.
"Yesterday morning they called me and told me that it would go out that night and in the end, he says he's staying, so he stays."
Pique said he only knew of Griezmann's decision on Thursday, revealing he immediately told Barca.
"I have not betrayed Barca and I'm surprised anyone would doubt my commitment to Barcelona because I've been a fan since birth," he said.
"I found out that Griezmann was staying at Atletico Madrid the same morning the report went out and I told people at the club."
The defender is a constant source of controversy and was once again in the headlines on Friday after claiming that Portugal star Cristiano Ronaldo was "prone to diving."