PC Gareth Phillips, 42, was left with potentially life-changing injuries - including a broken pelvis and head and abdominal injuries - after the incident in Moseley, Birmingham on Saturday.
He is in a stable condition in intensive care.
Mubashar Hussain, 29, was arrested in the Sparkbrook area of Birmingham and appeared in court today charged with attempted murder.
A West Midlands Police statement said: "It's alleged Mubashar Hussain hijacked a police car in Moorcroft Road, Moseley, at around 4.45pm and reversed into the officer, knocking him down, before driving over his body as he lay pinned under the vehicle."
PC Phillips' family said they were grateful for all the messages of support from the police and public.
"Gareth is still very poorly and faces a long battle to recover," they said in a statement.
"What happened on Saturday demonstrates the dangers that officers face every day."
PC Phillips has been with West Midlands Police for 17 years and worked as a traffic officer for the last 12 of those.
His father was also a police officer.
Mubashar Hussain also faces a number of other charges:
Causing serious injury by dangerous driving; dangerous driving; wounding another officer who suffered a cut arm; four counts of assaulting other police officers; driving while disqualified; driving without insurance; two car thefts and an aggravated vehicle taking relating to a police BMW.
The car theft charges relate to a Range Rover Sport that was taken on Saturday and another Range Rover stolen in Birmingham the previous day.
Another man, 24-year-old Ahsan Ghafoor, has been charged with the same car thefts, plus dangerous driving and other offences.
Both were remanded in custody and will appear at Birmingham Crown Court on 9 September.