Reference: Fataawa Ramadhaan – Volume 1, Page 166, Fatwa No.100
Fataawa Shaykh Ibn ‘Uthaymeen – Volume 1, Pages 476
Question: I intended to fast every Monday and Thursday but did not vow to do this. So is it obligatory upon me to fast [these two days of the week] all my life?
For example, if I fasted on a Thursday and then broke my fast [in the middle of the day], is it obligatory upon me to make up for this day?
Response: An act does not become incumbent just by having made an intention to do it. So if a person intended to fast every Monday and Thursday but then did not fast them, then there is no harm in this.
Likewise, if someone starts to fast and then breaks it, there is no harm in this. This is because it is not upon a person to complete a voluntary fast. Even if a person intended to give money in charity and thereby decided where and how much to give, then still it is not binding upon him to donate as he had intended, because in matters like these, mere intention does not make the act obligatory.
Based upon this, we would say to the questioner that it is not obligatory upon you to make up the fast that you broke on Thursday, and also it is not obligatory upon you to continue to fast every Monday and Thursday. However, if you were to do so, then it is good, since the fast of both Monday and Thursday is from the Sunnah.