Johnston, 19, claims the former vice presidential candidate has been part of an effort to keep Tripp away from Johnston by exerting influence on 19-year-old daughter Bristol, who is Tripp's mother.
"She has a big impact on what Bristol does and thinks," Johnston tells Entertainment Tonight. "Bristol looks up to Sarah. I don't know why they just don't want me in his life."
But Johnston insists he doesn't fear Palin as he gets ready to sue for joint custody. "She ain't nothing," he says. "Just because she ran for vice president and was the governor of Alaska, doesn't intimidate me. Sarah has no legal thing on Tripp, she's just the grandmother."
He adds: "By the end of the year we're gonna go to court for joint custody. I'm just done with it. When there's a kid involved you ain't walking away."
When Johnston announced his intention to sue for custody, Palin's camp did not respond to requests for comment. But her lawyer, Thomas Van Flein, told Britain's The Guardian on Oct. 30, "Other than noting that Tripp's father is always welcome to visit his son, we are unable to respond to these allegations as it is inappropriate to discuss child custody matters publicly."