I was thinking of the modern issue with the name 'Macedonia' being an object of contention between Greece and 'the former Yugoslav Republic of Macedonia' and was curious if it had come up with the name 'Mexico' in the past.
Did Mexico make any comment on the matter of naming New Mexico at all?
I'm not aware of any protest.
The name "New Mexico" predates the nation of Mexico. Santa Fe de Nuevo México was the name of the province back when it was still part of the Viceroyalty of New Spain. The early explorers thought they saw a lot of similarities between the pueblo dwellers of the upper Rio Grande and the Mexicas of the Valley of Mexico. So they named it New Mexico.
The name was thus in constant use from the late 1500s until the Mexican-American war by first the Spanish and then the Mexican governments. The US simply retained the pre-existing name, much like they did for Texas and California.