He also took aim at chief executive Omar Berrada, sporting director Dan Ashworth and technical director Jason Wilcox, who all arrived in the summer from Manchester City, Newcastle United and Southampton respectively.
'The new United hierarchy have made a deeply unimpressive start to their Old Trafford reign with one of their worst decisions being the most important any board can make,' Carragher wrote.
'Above all, you must ensure you have the right manager. Retaining Ten Hag is proving the catalyst for another wasted campaign and possibly another £200m down the drain.'
Erik ten Hag is under the microscope after Manchester United's poor start to the season
Ratcliffe was the key decision-maker behind handing Ten Hag a contract extension in the summer after the Dutchman guided the Red Devils to a shock FA Cup victory over Manchester City.
The United co-owner also sanctioned Ten Hag's £179million spending spree in the summer as Leny Yoro, Manuel Urgate, Matthijs de Ligt, Josua Zirkzee and Noussair Mazraoui all joined.
However, United sit 13th in the Premier League this season and have lost three of six games, with a 3-0 defeat to Tottenham last Sunday especially damaging.
As for some of the summer signings, most are yet to convince, with Yoro yet to feature in a competitive fixture after he was injured during pre-season.
Carragher felt it was a large sum of money to spend on players, especially since Ten Hag faces an uncertain future.
'And did no one raise the issue that if sticking with him backfired, his eventual replacement might not believe Matthijs de Ligt, Joshua Zirkzee and Manuel Ugarte are the right fit for them?' Carragher said.
MORE TO COME...