There is a new-ish law in California, which however does not cover your problem, it covers the infamous out-of-network surprise bill problem. There is nothing specific about doctor-patient contract, instead this falls under the scope of ordinary contract law. The legal question is whether you agreed to paying the charges. The exact details of who said and signed what matter. The written "confirmation" by the doctor that this will fall under the "routine checkup" category superficially looks like a good argument that you entered into this contract and would not be liable for the charges, but superficial appearances are not good enough. First, there is a point at which you sign a contract, and the wording of that contract contract is crucial. It no doubt commits you to paying the services, period, and does not allow the counter-argument "My insurance company is supposed to pay" (you have to sue the insurance company if that's where the problem is). Statements outside the written contract are very unlikely to override the written contract, especially if the written contract follows verbal or emailed suggestions.
The nature of your instruction to the doctor also matters, since "is covered by insurance" is a pretty loose instruction. My understanding of my own insurance policy is that there is an unavoidable deductible, and there are also things not at all covered, where is you spend money on those services, they don't even count towards the deductible. You may have meant "so that I will not have to pay anything at all", and he may have understood the instruction to mean "is in the set of things classified as 'covered' by the insurance policy". You describe this as "covered only after deductible is met", and if that's a quote from the insurance policy, means that the doctor cannot just look up coverage codes, he has to know about your prior medical costs for the year.
So it first depends on the exact language of your contract with the doctor, then on exactly what you said to the doctor (your condition for accepting the contract), and his exact response. He may well have included some disclaimer like "usually" whereby he makes no promises about insurance, and do you still agree to the procedure? That would be his primary legal leg, that he had disavowed your condition and you decided to proceed. The wording of his written response is what you should look most closely at.