Yes and no.
They were the first band ever to be described using the term 'heavy metal', so yes, they are literally the first ever heavy metal band.
However, since then the term has undergone a change to mean something entirely different, so no, by today's standards of what defines heavy metal they are not heavy metal, they are blues rock at best.
The real problem is defining the genre. A lot of the heaviest of heavy metal bands have elements of neoclassical rock to them, a style basically invented by Ritchie Blackmore, a classic rocker, and then popularised by Eddie Van Halen, about as close as you can get to glam rock while retaining a shred of dignity. The melodic solos of a lot of Metallica could have been taken from any place or time in music, including classical guitar, and so apparently the only thing that makes them metal is the fact that they have loud distorted amps, in which case Jimi Hendrix was metal because he had loud distorted amps.
Also, the genre of 'metal' has become too big to be sensibly applied anymore, a lot like the genre of 'rock' or 'pop.' There's just too much that fits into it.
Rock and Roll, Blues Rock, Heavy Rock, Soft Rock, Adult Oriented Rock, Pop Rock, Pomp Rock, Classic Rock, Glam Rock, Neoclassical Rock, Boulder, Pop, Power-Pop, Nineties Pop, British Pop, American Pop, Pop Idol, Soda Pop, Popular Music (the broadest term of all), Metal, Hair Metal, Heavy Metal, Death Metal, Screamo Metal, Speed Metal, Iron ore composite.
Basically the terms are meaningless because of how much overlap almost every band has nowadays. The Beatles' Helter Skelter actually has a lot of the hallmarks we'd associate with heavy metal today, while bands like Aerosmith cover everything from heavy metal to blues rock to pop rock depending on what sort of mood they're in. Just enjoy the music, don't try and label it.