The former professional cricketer player was involved in the accident while shooting the hit motoring show at the Top Gear test track at Dunsfold Aerodrome in Surrey on 13 December.
On Thursday, the corporation released a statement which said: “(BBC Studios) have sincerely apologised to Freddie and will continue to support him with his recovery.
“Under the circumstances, we feel it would be inappropriate to resume making series 34 of Top Gear at this time.
“We understand this will be disappointing for fans, but it is the right thing to do, and we’ll make a judgment about how best to continue later this year.
“This has also impacted the production team, who we continue to support. Finally, there will be a health and safety review of the show, in line with our procedures.”
The 45-year-old, who joined the programme in 2019, presents the show alongside Paddy McGuinness and Chris Harris.
Following the crash, Freddie’s son gave an update on his dad’s condition, stating he was “lucky to be alive”.
Speaking to the Daily Mail, 16-year-old Corey said: “It was a pretty nasty crash. It is shocking. We are all shocked but just hope he’s going to be OK.”
Freddie, who was appointed England cricket captain twice in his career, retired from the sport in 2010, briefly returning to play for Lancashire four years later.
Following his successful sporting career, he became a familiar face on television, becoming a team captain on the sports quiz A League Of Their Own and later winning the Australian version of I’m A Celebrity… Get Me Out Of Here! in 2015.