Here's why:
* Linking verbs connect the subject of a sentence to a predicate nominative or predicate adjective. They don't express an action.
* Predicate nominatives are nouns or pronouns that rename the subject.
* Predicate adjectives describe the subject.
In this case, "is" links the subject "he" to the predicate adjective "right." It tells us what kind of person "he" is in relation to the subject matter.