IMO I would have to say that the ninja in the Naruto anime do cast spells because they do rely spell components much like a wizard does. Their spells or Jutsu (as they are called in the anime) do require a Somatic component and a Verbal component. The former takes the form of
Hand Seal Each of these basic hand seals are modeled after the 12 animals of the Eastern Zodiac. There are other hand seals besides these 12 in the anime. As for the latter, it often takes the name of the elemental style (Wind Style, Water Style, etc.) and the name of the Jutsu itself. "Water Style, Water Dragon Jutsu!"
Their Jutsu doesn't rely on a Material component. Instead they're powered by Chakra (or Ki), which is a mix of physical energy and spiritual energy generated by a ninja. It comes three different varieties- Ninjutsu (Ninja Techniques), Genjutsu (Mind Techniques) and Taijitsu (Body Techniques)
Naruto: Shinobi (5e Class) - D&D Wiki Check out this homebrew for 5e.