It is understood Carver has agreed a new improved contract at the Championship club, who are fighting a relegation battle.
He said he had rejected Burnley's offer out of loyalty to Mariner.
Carver told Radio Five Live: "The club's board has a five-year plan and I want to be part of that.
"I felt I had to be loyal to Paul for giving me this opportunity. In the short-term we have a lot of hard work to keep Plymouth in the Championship."
Laws was appointed by Burnley on Wednesday as Owen Coyle's successor on a two-and-a-half-year deal.
His backroom staff already includes former Scunthorpe and Sheffield Wednesday assistant Russ Wilcox, while 38-year-old captain Graham Alexander will take up a new role as player-coach.