You piqued my interest with this question. According to the agreement itself (below), Article 13 clearly indicates that Migración should accept an apostille from any U.S. state as follows:
Article 13
Any State may, at the time of signature, ratification or accession, declare that the present Convention shall extend to all the territories for the international relations of which it is responsible, or to one or more of them. Such a declaration shall take effect on the date of entry into force of the Convention for the State concerned.
At any time thereafter, such extensions shall be notified to the Ministry of Foreign Affairs of the Netherlands.
When the declaration of extension is made by a State which has signed and ratified, the Convention shall enter into force for the territories concerned in accordance with Article 11. When the declaration of extension is made by a State which has acceded, the Convention shall enter into force for the territories concerned in accordance with Article 12
Obviously, the United States as the signatory has extended the authority of the convention to all 50 U.S. states, and I imagine Puerto Rico, Guam, American Samoa, USVI .... etc as well.
However, I would bet dollars to donuts that the reflexive response from the indolent and intellectually incurious will be to refuse out of hand your New York state apostille. I doubt that many of them know, or care, that an apostille almost always merely confirms just the signature of the document, verifies the authenticity of the seal / stamp and affirms the capacity of the person signing the document. U.S. Dept. of State Apostilles do not directly verify the contents of the letter. It does so indirectly by verifying (1.) the signature, (2.) the seal / stamp and (3) the capacity of the person who signed the document.
I would double the bet by asserting that 85% of Migración employees never read the convention itself. So, while I believe that you are legally entitled to submit and have accepted your NY state apostille, I think that you are entering a Franz Kafka type of Idiocracy struggle session with federal workers. I wish you all the best. Please let us know how it works out.