Is it obligatory to complete the fasting of a month vowed to Allah the Almighty after breaking the fast unintentionally on the first day, or is it obligatory to re-fast that day?

Your fast is not broken by unintentionally breaking it. You must complete the fast, and you do not have to make up another day, according to the Prophet's (peace be upon him) saying: "Whoever forgets while he is fasting and eats or drinks, let him complete his fast, for it was Allah who fed him and gave him drink." As for one who vows to fast an unspecified month, most jurists hold that consecutive fasting is not obligatory unless he intends it or stipulates it at the time of the vow. This differs from the Hanbali school, which believes that consecutive fasting is necessary.