As far as I know, it is. We try to make French words sound French.

That would be "herbe".

What part of the US pronounces the H in herb?

I don't know that it's a regional variation, but about half of the people in my family pronounce the "h", including me most of the time. My sense is almost that dropping the "h" is an "educated" thing, and those who don't drop it are unconsciously tweaking the whole concept.

What part of the country do you live in if you don't mind me asking?

I'm just curious so that I can try to imagine the accent.

I live in the southeast, we say things like reckon, yall, and wander, but "erb" is still the accepted way to pronounce herb 'round here.