I believe more information is necessary to give you an accurate information. What about the child's other parent? Are they a United States citizen? Was your child already admitted to the United States as an LPR when you naturalized? Citizenship acquisition and conveyance can be a bit complicated, so it may be easier to call an attorney. The attorney may be able to give you an answer after a brief consultation.
You may be able to convey US citizen to your child who was born after you acquired US citizenship through naturalization if the child was under the age of 18 at the time you naturalized. In addition, the child must be lawfully in the US residing as in permanent residency status. This means that you will need to sponsor your child for an immigrant visa and then once the child enters as a permanent resident they may immediately apply for a US passport.
Please note that by answering this question an attorney client relationship is not created. My answer is general information based on the facts provided in the question. Detailed and accurate advice may only be given once a lawyer is able to meet with a person to assess the totality of a person's circumstances.
You cannot get him a US passport at the US embassy, as you were not a US citizen when he was born. He needs to be in the US in order to naturalize.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.