"Watching soccer games" is the subject but the teacher argues that it is the word"watching"
Yes, the teacher is right. The gerund phrase watching soccer games acts as the subject, and watching is the head word (a verbal noun). It is considered to be singular (uncountable), and it takes a singular verb (makes). The correct sentence is:
Watching soccer games makes me happy.
Soccer games is the object constituent of the gerund within the phrase.