Ground level falls are considered minor trauma with exceptions of the geriatric population which constitute the largest amount of fractures. The information you provided regarding the fall was very limited. So I'm assuming he didn't take a fall down a flight of stair, off a balcony, or from a height 3 times greater than standing.
The paramedic assisting your father probably took into account such information aka obvious mechanism of injury, how he landed when he fell, direct trauma to head and torso sustained, as well as many other things only experienced paramedics are aware of. In addition to the basics like loss of consciousness, nausea, vomiting, sensory deficits, limitations of ROM, and absence of spontaneous respiration which are all common in primary spinal cord injury and most nurses know.
Sometimes rigid C-collars and backboards cannot be applied without the risk of increasing injury. These devices may actually do more harm than good especially in the elderly e.g. spinal arthritis, kyphosis, hyperlordosis, risk possible increase of intracranial pressure if a head injury was sustained, airway management complexities etc.
Secondary spinal cord injury often occurs en route due to swelling, bleeding, and tissue ischemia so time is very much a factor. Again, I don't have enough information regarding your father's condition but the medic has less than a 'Golden Hour' to decide whether to stay and play or load and go.
It is doubtful the ED would place him in a collar unless something occurred to lead them to believe otherwise e.g. failed neuro check, swelling or bulges on the C-Spine. A CT scan of the C-spine may be appropriate but that depends on the protocol and the ordering ED doctor.
Professional courtesy goes right out the door when you start impeding care. The last thing the ED staff needs, is someone second guessing what they do every day. Even if you're a trauma nurse, you know there are ways to handle things if you think someone is being medically negligent. I'd urge you to consider, what would you do if someone came onto your floor and started making the same demands before you continue your course of action.
What was the final outcome? Was a fx to c-spine sustained?