First, I am not alone in classifying [h] as a fricative. That is how it is categorized on the IPA Chart; that is how Cruttenden, Roach, Collins, Ashby & Maidment and other respected writers on English phonetics categorize it.
Secondly, however, it is clear that [h] is different from the other English voiceless fricatives [f, θ, s, ʃ], in that it does not involve a constriction within the mouth cavity. Conventionally we classify it as a glottal fricative, but the constriction giving rise to the turbulence that we hear as friction may be not so much located at the glottis itself as distributed throughout the whole of the upper vocal tract.
That is why Ladefoged and Maddison in The Sounds of the World’s Languages (Blackwell 1996), at the beginning of their chapter on fricatives, comment
Forms of h, ɦ in which a turbulent airstream is produced at the glottis are also sometimes classed as fricatives […], but it is more appropriate to consider them in the chapter on vowels.At the end of the vowels chapter they mention the possible description of [h] as the voiceless counterpart of the vowel that follows.
In such cases it is more appropriate to regard h and ɦ as segments that have only a laryngeal specification, and are unmarked for all other features.But in some languages, including Hebrew and Arabic, a glottal constriction is observable during the production of these sounds.
One problem with classifying [h] as an approximant is that voiceless approximants are by definition inaudible. (Or by one definition, at least. Approximants used to be known as “frictionless continuants”.) If there’s no friction and no voicing, there’s nothing to hear. Anything you can hear during a voiceless [h] must be some sort of weak friction, resulting from some sort of weak turbulence, which means that [h] is some sort of weak fricative — but still a fricative.
The English /h/ phoneme does not behave like a vowel. We say a house, not *an house. We say ðə house, not ði house. You get a linking r in you’re out, but not in your house — except among non-standard speakers who drop h.
The tradition in generative phonology is to class [h] as a glide, along with [j] and [w]. That’s fine phonologically, but not very helpful phonetically.
For practical teaching, it’s convenient to call [h] a fricative. But you do have to emphasize that there is no friction at the uvular or velar place (of the sort you get in [x] and [χ]). Many EFL learners can be helped by thinking of [h] as just a voiceless onset to the following vowel.
Advanced students can be asked to write an essay on the problem of defining the terms fricative and approximant.