This is a rather old joke but I got curious and checked the language origin of both words. As you can see from the entries taken from www.word-origins.com, one does not contain the other, and your friend can rest easy in knowing that the word diet will not do him any harm. His current mode of eating, though, may do that, and if that is so he better change to a more healthful one. And then he can call it whatever he wants 🙂
Diet comes, via Old French diete and Latin diaeta, from Greek d├¡aita ÔÇÿmode of life’. This was used by medical writers, such as Hippocrates, in the specific sense ÔÇÿprescribed mode of life’, and hence ÔÇÿprescribed regimen of food’. It has been speculated that Latin diaeta, presumably in the yet further restricted sense ÔÇÿday’s allowance of food’, came to be associated with Latin di─ôs ÔÇÿday’. This gave rise to medieval Latin di─ôta ÔÇÿday’s journey’, ÔÇÿday’s work’, etc, hence ÔÇÿday appointed for a meeting’, and thus ÔÇÿmeeting (of legislators)’. English acquired this word (coming orthographically full circle as diet) in the 15th century, but it is now mainly used for referring to various foreign legislatures.
English has two distinct words die. The noun, ÔÇÿcube marked with numbers’, is now more familiar in its plural form (see (dice)). The verb, ÔÇÿstop living’ (12th c.), was probably borrowed from Old Norse deyja ÔÇÿdie’. This, like English dead and death, goes back ultimately to an Indo-European base *dheu-, which some have linked with Greek th├ínatos ÔÇÿdead’. It may seem strange at first sight that English should have borrowed a verb for such a basic concept as ÔÇÿdying’ (although some have speculated that a native Old English verb *d─½egan or *d─ôgan did exist), but in fact it is a not uncommon phenomenon for ÔÇÿdie’ verbs to change their meaning euphemistically, and therefore to need replacing by new verbs. In the case of the Old English verbs for ÔÇÿdie’, steorfan survives as starve and sweltan in its derivative swelter, while cwelan is represented by the related cwellan ÔÇÿkill’, which has come down to us as quell.