What is the ruling on vowing to name a newborn after one of the Beautiful Names of Allah, despite opposition from family, and is expiation obligatory if the vow is not fulfilled?

A vow is not established by self-talk or intention; rather, it requires verbal utterance. If you did not utter the vow, then nothing is incumbent upon you. However, if you uttered it secretly and the condition it was made upon was met, then you must fulfill it. As for naming a child with the names of Allah, its details can be found in specialized fatwas.