All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
A vow is an offering that a person commits himself to present to Allah if a given good thing happens to him or a given bad thing doesn`t hit him, and it isn`t obligatory in Sharia. However, if the conditions of the vow are met, then it becomes binding because Allah The Almighty said: "Then let them complete the rites prescribed for them, perform their vows.."{Al-Hajj/29}. Moreover, the Prophet(PBUH) said: "He who vows to obey Allah, should obey Him. But he who vows to disobey Allah, should not disobey Him."{Bukhari}.
Moreover, if a person vows to fast a particular month, then he is obliged to fast all of it.
However, if he missed one day-whether with a valid excuse or not-then it is mandatory that he makes up for that day. Al-Imam An-Nawawi said(What means): "If he vowed to fast a certain month, such as Rajab or Shabaan, but missed one day, then he is obliged to make up only for that day. Similarly, if he vowed to fast for one month without specifying which month, and missed one day, then he is obliged to make up only for that day."{Rawdat At-Talibeen, 3/310}. And Allah knows best.