Maurice Hammond's aircraft came down at Hardwick airfield in Norfolk, near Bungay, and his unnamed passenger, aged in his 80s, died at the scene.
Mr Hammond was flown to the Norfolk and Norwich University Hospital, where he remains "sedated, to aid his recovery".
Family spokesman Nik Coleman said he was due to undergo further tests.
The pilot was wearing a helmet that gave him a "very high degree of protection", said Mr Coleman in response to "speculation online".
"His family have visited with him [sic] and are encouraged by his progress," he added.
More on this and other news from Norfolk
Mr Hammond, from Eye in Suffolk, is said to be one of the UK's top aviators of the American plane, an expert on plane restoration and has appeared on a Discovery Channel series, Plane Resurrection.
The P-51 D Model Mustang - one of two Mustangs owned by airshow regular Mr Hammond - was on fire when emergency services arrived at the site, about 10 miles south of Norwich.
A witness reported he saw it approach the runway normally but it then disappeared behind trees and he heard a small explosion.
A spokesman for the Air Accidents Investigation Branch (AAIB) said it was due to remove the plane from the crash site later, in an operation that could last until the evening.
"It will be brought back to our headquarters in Farnborough and investigations will continue here," he said.
It could be months before the inquiry is completed, he added.
Norfolk Police said road closures remained in place.
The family of one of the UK's top Mustang pilots, whose vintage aircraft crashed on Sunday killing a passenger, has been "encouraged by his progress".