Swann, who retired from cricket during England's humiliating series whitewash in the last Ashes tour of Australia, called for former team-mate Cook to cease his involvement in the limited-overs game to allow for a more exciting batting line-up.
The former off-spinner has criticised England's approach to limited-overs cricket as outdated and does not believe they have any chance of winning the World Cup for the first time next year.
However, his comments drew the ire of Cook, with the Essex opener expressing his disappointment at the remarks, given the pair's friendship.
Yet Cook's response has not altered the stance of Swann, who repeated his plea for the 29-year-old to stand aside.
Swann wrote in his column for The Sun: "Real mates are honest with each other. I'm no longer in the England dressing room and it is my job now as a pundit to give my honest thoughts.
"I think being one-day captain is a poisoned chalice for him. He just doesn't need the job. The cricket England are playing in 50-over cricket is outdated and I fear it will be a painful winter for the team.
"I don't want him to be part of it. This is not a witch-hunt against 'Cookie' and I'm still a huge supporter of his captaincy in Test cricket. But, in one-day matches, I want England to play the exciting cricket they promised six months ago when Peter Moores took over as head coach.
"We're happy to tootle along in a two-litre diesel in a Formula One race.
"England must change their strategy to have any chance of winning the World Cup.
"I'm 100 per cent behind him as Test captain. But, in one-day cricket, you should pick your 11 best players and choose your captain from there and I don't think he is one of the best 11 players."