In short, the answer to the question is no. The confusion seems to come from a conflation of distinctions between the syntactic passive voice and the semantically distinguished passive verb.
Syntactically, the passive voice distinction is only relevant to sentences with a transitive or ditransitive verb and a direct object verb complement in their active version. The passive voice changes the verb's form (be + past participle), the active form's DO becomes the patient/subject, and the verb's agent is generally identified in the complement. (With active voice ditransitive verbs, either object can become the patient subject of the passive voice.)
Semantically, verbs are distinguished as passive or active based on the thematic role of the subject (agent/patient) in relation to the verb, without regard to its transitivity. If the subject is the patient of the verb, the verb is considered passive. However, whether a verb is semantically passive or not does not bear on its transitivity; some passive verbs can take the passive voice (can be used transitively) and some cannot.
So there is a syntactic distinction based solely on the use of transitive verbs in the active voice, which allow the verb to be rendered in the passive voice. The semantic distinction based on theme (a passive verb) is not related to voice. Sentences in the passive voice use a syntactically passive verb form (be + past participle), that is also semantically passive.
To say that a sentence using a passive verb is in the passive voice is incorrect (See here). It is equally incorrect to say that the active voice requires a transitive verb. Any sentence not in the passive voice is in the active voice.
The sample sentences in the original post use a passive verb and are properly defined as being in the active voice. The passive voice always changes the verb's conjugation (be + past part.) and requires a transitive construction in the active voice.
I belong to a gym. (active voice/ intransitive passive verb)
This sentence cannot be made passive. 'Belong' is always intransitive.
She is a good teacher. (active voice/ link verb)
Link verbs cannot be made passive.
The grapes need eating. (active voice/ transitive passive verb)
This sentence can be made passive: 'Eating is needed by the grapes.'
The patient underwent surgery this morning. (active voice/ transitive passive verb)
This sentence can be made passive: 'Surgery was undergone by the patient this morning.'