Is an object of a verb a complement of a verb or of a verb phrase (a predicate)?

The object of a verb is included in the verb phrase headed by the verb.

A clause may be analyzed at different levels and it is important to differentiate function and category of the elements.

For example I love you deeply.

At the most general level, it is typically split into predicate love you deeply and subject I.

At a more detailed level we have complements I and you, predicator love and adjunct deeply. Both object and subject are categories of complement as they are permitted - or licensed - by the predicator. The subject I is an external complement and the object you an internal complement. The adjunct deeply is part of the verb phrase in this case as it is a degree modifier. So we have VP as the category of love you deeply which happens to be the same as the predicate in the functional analysis at the clause level in this case.

As for complements completing the meaning of a sentence, complements may be required or optional: required for verbs like become, and optional for verbs like ate. In this view of complements, they do not seem to complete the meaning of the sentence where optional, but rather clarify or modify it where they are found.

* The house became.

I ate.

The first is ungrammatical and requires a predicative complement. The second is grammatical as it is, but could take an object as a complement to make clear what was eaten.

As to the meaning of expression, this is not a term that is typically used in grammatical analysis.

Taking expression to mean a word or group of words used in a particular situation, the expression in (a) is I love you. In grammatical terms, this is a clause with predicator love and complements I (subject) and you (object).

If complete the meaning is the same idea as make grammatically acceptable, then we see that you is required in this sentence as love is a verb that requires an object. Love by itself has meaning, but the clause I love would not generally be considered good English. The fault clearly lies in the predicate, not the subject, so we may say that the predicate is incomplete as it is lacking the required object. The addition of you would make the predicate grammatically acceptable or complete and at the same time make the whole clause grammatically acceptable and hence complete.