Simple answer: no. There is no phonological rule that I’m aware of that requires ‘doubling’ any final glides (or making a diphthongal offglide out of a long monophthong) in any variety of English.
It’s certainly true, however, that many speakers (not just of American English, but also of other dialects) add a brief [ə] between any high vowel and an [ɫ], but this is not a mandatory process as much as it is quite a natural one.
The velarised alveolar lateral approximant [ɫ] is quite a complex sound in that it has two places of articulation, both defined by parts of the tongue: the tip (or blade) and the root are both raised towards the top of the mouth. In order to retain maximally clear differentiation, the central part of the dorsum of the tongue is lowered as far as it can comfortably—otherwise, you’d just get one long place of articulation all along the dorsum, which is audibly very different. You can imagine the shape of the tongue for saying [ɫ] as more or less shaped like this:
Conversely, the high vowels have only a single place of articulation, which is defined by the central part of the dorsum (in the case of [uː], the posterior part of the dorsum, but usually still a place slightly anterior to the root) being raised towards various points on the hard palate, while the tip/blade and root are in a neutral or lowered position. You can imagine the shape of the tongue for high vowels more or less like this:
In other words, when moving from a high vowel to a [ɫ], the tongue needs to transition from
∪. Since both sounds are voiced, you’re quite likely to hear the entire transition; and obviously, the midpoint when moving from
—, a neutral, flat tongue in the middle of the mouth—or in different terms, the tongue position needed to produce [ə].
So it’s quite natural to insert a [ə] between a high vowel and a [ɫ], and English is not the only language that does this.1
Now, if you add a schwa to a long vowel or diphthong in a closed syllable, you very easily end up with a glide (either the last part of the original diphthong or a strengthening of the quality of the long vowel) standing more or less consonantally between the two vowels, even if this glide may be more weakly pronounced than a real, phonemic glide. So [maɪɫ] becomes [maɪəɫ] becomes [maɪ.(j)əɫ], and [fiːɫ] becomes [fiːəɫ] becomes [fiː(j)əɫ]. My immediate perception of this, however, is that while some speakers, particularly in the southern parts of the US, do quite audibly strengthen this glide and pronounce it as a very audible, consonantal glide, there are many speakers who do add a [ə] but do not significantly pronounce any offglide-like sound in the preceding vowel or diphthong.
Acoustically, the impression may often be that a glide is added, but I would say the glide is a secondary result of the inserted [ə], which is far more common—though still perhaps not quite universal.
Note that this [ə] is much less likely to ever occur in speakers who do not use velarised [ɫ] post-vocalically, but have non-velarised [l] everywhere. This includes some dialects of British English and most importantly nearly all dialects of Irish English.
For more details than you ever knew you cared for about schwa epenthesis before liquids—though mainly dealing with /r/ rather than /l/—I refer you to this (highly technical) article by Martin Krämer from the University of Tromsø (Norway).
1 Off the top of my head, I can think of Irish and Scottish that do the same, as well as (at least) German and Danish, and partially also French, for the comparable sounds [ʀ ʁ]. I’m sure there are many other similar examples.