Large company: Made up of organizations or departments?

In the context of Apple, I would agree that departments is the better word. HR is a department, as is Support.

Generally speaking, an entity can comprise several organizations, but in that case I would call that entity itself an organization (or, in case the entity is commercial, corporation). As a simple example, the UN is an organization, but so are its agencies such as the UNTSO and the WHO.

Looking at the Merriam-Webster definitions, I see that organization is an "association, society", while department is "a major division of a business". HR, Support, and iOS Development are certainly major divisions of Apple, but they can't be labeled as societies or associations in their own right.

I think rather than using organization or department, IMHO the right term is functional groups.

A company usually has departments; a large company may have divisions (which in turn have departments within them).

I think "department" and "division" have a stronger connotation of hierarchy than "organization" does. Organizations can be looser associations, while the engineering department has a definite reporting chain (to the CTO or whomever).