Carver said in the Yorkshire Evening Post: "Leeds had the decency to allow me to talk to Carlisle and I'm delighted with the way the two clubs have acted.
"I had to make a decision and on this occasion it's to stay at Leeds. I enjoyed last season, although it was disappointing how it ended, and we still have a job to do. I want to be a part of that."
Carver was believed to be the first choice of Brunton Park owner Fred Story but the former Newcastle assistant boss has been at the forefront of Blackwell's coaching staff for the last 12 months, and his contract is due to run until the end of next season.
Leeds are still to confirm whether he will be promoted to assistant boss following Sam Ellis' departure last month.
Blackwell added: "I'm always willing to give an opportunity to my staff to speak to someone else, but I'm delighted John sees his future at Leeds United."