Upvote:3
From personal experience (although not with the US), I had a current Schengen visa in one of my passports when I got my US citizenship. On my next trip to Europe I used my US passport, despite having another passport with a still valid visa which was valid for the purpose of the visit. Nobody asked any questions.
Now regarding the USA, I have found no specific information about it in FAMs or on DoS website about that. Thus my answer would be more of a speculation conclusion, and it would be "no, you would not be required to use a specific passport to enter the USA" simply because you have a visa in that passport. Consider the typical case where the passport with the visa is no longer in your possession because it has been replaced/lost/damaged.
Interestingly enough, the US visa application form doesn't even ask about all your nationalities.
Upvote:3
If you have two nationalities, you may enter the US using either one you wish.
If you enter using a VWP-eligible nationality, you must have an ESTA, and if you enter using a nationality that requires a visa, you must have a visa.
I have dual citizenship from both a VWP country and a non-VWP country?
If you are flying under the terms of the VWP, with your VWP passport, you must apply for ESTA. If you are using your non-VWP country's passport, you will require a visa and therefore ESTA does not apply to you.