Verbs of perception like see, hear, watch, feel take both -ing-form and bare infinitival† clauses as complements, but there is a slight difference of aspect between them:
The infinitival complement implies that what is perceived is a completed action.
He watched me play means that he watched until I was finished playing.
The -ing-form complement implies that the action continues while it is perceived, but is not necessarily finished during that period.
He watched me playing means that he watched for some time while I played, but implicates‡ that he stopped some time before I finished.
Your example I grew up watch his movies, however, is ungrammatical. Grow up is not a verb of perception and does not take bare infinitival complements. It takes marked infinitival complements, which describe an outcome of growing up, and subjectless -ing-form adjuncts, which describe actions and states of the person growing up during that period.
I grew up to watch his movies means that you watched his movies after you grew up—perhaps you knew him when you were a child but didn't realize he was a famous actor until later.
I grew up watching his movies means that you watched his movies while you were growing up.
† A bare infinitival clause is one headed by an infinitive which is not marked with to.
‡ Implicates is a technical term for having a 'default' meaning which may in fact be contradicted.