I went to the physician in California, United States. The secretary orally told me that the appointment was covered by my US health insurance, but it was not. Can I refuse to pay on the grounds that the secretary told me I wouldn't have to pay anything?
You can refuse to pay.
The doctor can either accept your refusal or pursue the debt in court. If he pursues the debt you will probably lose.
The Common Law position
Your contract with the doctor was for him to do whatever he did and you to pay for it. Your contract with your insurer is for you to pay the premium and them to reimburse you for whatever they cover.
The fundamental question is why you were taking the doctor's (via his secretary) advice on your contract with your insurer? The advice was wrong, however, it is difficult to see that there is a case for negligent misstatement; you would have difficulty showing there was a duty of care and even if you did showing what damage flowed from it since it is quite likely that you would have had the procedure notwithstanding the absence of cover, unless it was purely cosmetic.
The Consumer Law position
California probably has consumer protection laws regarding misleading and deceptive conduct - I have no idea what they are. If this is so then your doctor's statement was misleading - consequences may flow from this.