Is what the questioner said considered an oath, and what are the consequences if it is?

The speaker's statement, "I will not call so-and-so (female)," is not an oath. An oath is made by Allah, or by one of His names, or one of His attributes, or by the specific phrases to which Islamic law has assigned the ruling of an oath. Therefore, nothing results from this statement. He should strive for reconciliation, maintaining kinship ties, and ensuring the child is reunited with its mother. He should also seek the help of virtuous and wise people to prevent injustice from befalling the child and its mother.