Nationality on my marriage certificate

Clearly the VN officials don't really understand the ins and outs of US green cards and citizenship procedures.  Unfortunately your path of least resistance may be to do as they wish and go to the US Consulate.  Although you wrote US Embassy, you are posting on the Saigon forum so I wrote Consulate.  Hopefully they have encountered this problem before and know how to handle it.  I hope you did not avail yourself of the option of a free name change when you naturalized.  If so, that may complicate things.

Looking at my step-son's birth certificate, I see that there are separate entries for all three individuals (two parents and the child) for both Ethnicity (dân tộc) and Nationality (quốc tịch.)  In your case I expect these entries should say Kinh and Hoa Kỳ.

Congratulations!!!

You have Vietnamese and US citizenship.  You are a dual citizen.

You have two choices.

1) Write Vietnamese on the birth certificate, and then go to the US Consulate to file the CRBA and apply for his US passport.

2) Go to the US Consulate and make the affidavit that you are the same person on the Vietnamese marriage certificate and the US passport.  Then go back and put American on your son's birth certificate.

https://vn.usembassy.gov/u-s-citizen-se … -services/

Hello, thank a lot for replying! I really appreciate it. However, I am kind of afraid that putting my nationality as a Vietnamese would affect my son's CRBA, would it? I am aware that my son's nationality is okay whether his nationality is Vietnamese or American.

No, putting Vietnamese will not affect your son's CRBA.   Your US passport will prove your US citizenship for the CRBA.  If you are afraid, then go get the affidavit the VN birth registrar wants.