Does my father have a case if he was being transported from the hospital to a long term care facility when a private ambulance company dropped him from the gurney?

The problem is, there are two elements to a successful personal injury case.
The first is liability, or fault: the person or business (I will call them "defendant" from here on) you sue must have done something intentially to injure you, or else injured you through negligence, or unreasonable carelessness. There seems to be liability in this case; it seems negligent to drop a patient from or off of a gurney.
But the second element is "damages," or what you can sue for. You can only sue for the sum of:
1) The out-of-pocket (not paid by insurance, medicare, or medicaid) medical costs the careless act caused you.
2) Lost wages and reduced future earning potential, if any. 
3) For injures causing *significant* disability or life impairment lasting typically months, years, or more, some amount for "pain and suffering"-- but the amount you can get will be minor until you start geting to permanent damage to a joint, impairment of vision, prominent facial scarring, constant lifelong pain or discomfort, etc.
Some stiches on the top of his head and pain for a few weeks will not generate much, if anything, in terms of a pain and suffering award. So unless your father incurred large medical bills specifically from the fall or lost significant amounts of earnings due to the injury, there is no point in suing--you will spend more on the lawsuit (even without a lawyer, since you'll need to hire a doctor or other medical person to testify about his injuries unless you're only suing for easily proven medical costs) than you will get back.