Weeping Cross Health Centre in Stafford has been closed since Monday after a silver Citroen caused 'structural damage'.
It is believed the incident itself took place on Sunday, meaning a busy Monday of appointments could not be fulfilled.
The centre, on Bodmin Avenue, is usually open Monday to Friday, 8am to 6.30pm, but it is unclear when it will be safe to for its doors to be reopened.
The centre has sustained structural damage and the utilities have been turned off.
Councillor Ann Edgeller, Stafford South East County Councillor said: "I first heard about it on Sunday.
"My first initial contact was from a community group post on Facebook that said an incident had happened.
"People were calling me to ask whether it would be open on Monday as they had appointments.
"From that, I phoned the practice manager to find out and he said there was structural damage inside and they couldn’t open at all.
"They are not sure when they will be able to reopen, but we do need it reopened as soon as possible."
A joint statement by Councillor Ann Edgeller and Councillor Marnie Phillips said: "Unfortunately a car crashed into Weeping Cross Surgery today and the building has had extensive damage.
"The driver was badly shaken but thankfully didn’t receive any injuries, so we understand.
"The surgery will be closed to patients due to the structural damage sustained, along with the utilities being off.
"It’s our understanding that all patients with early appointments tomorrow have been informed.
"All other patients will be contacted as necessary, once the extent of the damage has been assessed."