The LCR Honda rider, 31, crashed at Turn 10 during Saturday's practice session because of oil on the track.
Crutchlow said he was advised by MotoGP medical director Angel Charte to pull out of the qualifying session.
"I am a racer and I want to race," said the Briton after earning his grid place for Sunday's race in Brno.
"I respect Dr Charte and all the doctors in the clinic a lot, but the decision was mine and that's why they got angry.
"Now I have to go back to the hospital because they saw there was something [wrong] with the vertebrae."
Crutchlow went for further examinations after qualifying, but the hospital checks did not reveal any significant injury and he is expected to be fit to race on Sunday.
Meanwhile Honda's world championship leader Marc Marquez will start on pole for the fourth time this season.
The three-time world MotoGP champion finished ahead of Italian veteran Valentino Rossi, of Yamaha, and Spain's Dani Pedrosa, of Honda.
Cal Crutchlow says he defied medical advice to qualify fifth at MotoGP's Czech round amid concerns he had suffered a back injury in a crash.