What is the ruling on not fulfilling a vow I made to fast a day for Allah and not to love anyone until engagement, given that I fasted the day but was unable to fulfill the second part of the vow?

It is not advisable to make a vow (nadhr) in the first place. If it is a vow of obedience, it must be fulfilled. If it is conditional upon the attainment of a blessing or the warding off of a misfortune, it must be fulfilled upon the fulfillment of the condition.

The questioner must fulfill the vow of fasting for one day if the matter upon which the vow was conditioned occurs.

As for a vow to stop loving someone, if what is meant by it is to cease establishing an unlawful relationship, then this is a vow to abandon sin, and abandoning sin is obligatory. Some scholars believe that this vow is not valid, and repentance from committing the sin is required. Others believe it is valid, and an expiation for breaking an oath (kaffarat yameen) is required along with repentance. This latter opinion is the more cautious one.