In New York the answer is yes, you can.
Want to know why?
It's because the issue of whether he charged you and whether you paid are distinct and different from whether he treated you appropriately.
Let me put it a different way.
Your doctor has an obligation to treat you in accordance with good medical practice.
If he violates the basic standards of medical care and his carelessness was a cause of your injury, you can sue him for medical malpractice...assuming of course a medical expert confirms that you have a valid case.
Whether your doctor charged you or forgot to charge you has NOTHING to do with whether he was careless on a particular date and time.
Let's take this hypothetical example and see what happens...
What if your doctor charged you for a visit and you failed to pay.
Let's say you tried to schedule a follow up appointment.
When you called his office, his receptionist refused to schedule a new appointment until you paid your bill in full.
You are unable to pay, for whatever reason.
At your last visit, your doctor told you to follow up with him as he wanted to follow your suspicious findings.
You're now calling to schedule that follow up.
You tell the receptionist this.
She doesn't care.
She's been told to demand payment in full.
Otherwise, her marching orders are no further appointments.
That creates a problem.
A legal problem.
An ethical problem.
Is it legal to prevent an existing patient from returning to see your doctor if payment has not been made?
What if this delay in paying causes a signficant delay in having the patient seen for diagnosis and treatment?
Would the doctor be liable for the delay in diagnosis and treatment?
Would his office be liable?
Would the patient be responsible for causing or contributing to her own delay in getting timely medical care and treatment?
There are no easy answers to those questions.
The real answer is "It depends."
On many factors.
If the doctor told the patient to return to closely follow her condition because he was concerned about it, and his office refused to schedule a new visit, they may bear SOME responsibility if the patient's diagnosis and treatment were delayed causing her injury.
If the doctor did not give the patient alternatives to getting appropriate medical care in light of the payment problem, the doctor and his office may bear SOME responsibility if the patient's diagnosis and treatment were delayed causing her injury. That might be a form of abandoning a patient.
Some lawyers might say this is 'constructive abandonment'.
You told the patient to do something and then put up a barrier to having it done.
That's not a good situation for the doctor or the patient.
Could the patient be responsible for delay if she could no longer afford to pay?
The answer is maybe, but if the explanation justifies why she could not pay, the jury may feel for her and understand her predicament.
The bottom line is that whether you've paid or not has NOTHING to do with whether your doctor treated you in accordance with good medical care.
Let's see a doctor and his attorney try and argue to a jury during trial that the reason he's not responsible for your injuries is because you didn't pay your co-pay.