
From what I've seen, the one that comes close is Shin Godzilla. Unfortunately, I can't think of another that embraces those same horror undertones from the original, at least not to the degree of SG.
"What’s so great about discovery? It’s a violent, penetrative act that scars what it explores. What you call discovery, I call the rape of the natural world."
— Dr. Ian Malcolm, Jurassic Park (1993)

For Godzilla to become a horror movie, that would mean turning Godzilla to the dark side. No, I don't think that will happen.

I'm hesitant on calling any Godzilla film (barring 1954) a horror film. Although movies like Shin, GMK, 84, Bio, and some others do most certainly contain horror elements.