This is actually referring to different species of grapes, and has nothing to do with winemaking technique or prowess.
Virtually every type of "quality wine" that is consumed in the world comes from the old world species vitis vinifera. This includes basically every variety you've heard of: Chardonnay, Cabernet Sauvignon, Pinot Noir, Syrah, Merlot, just to name a few French varieties (but the same applies to Italian, Spanish, German, etc. wines). Any American winemakers making those kinds of wines (and I agree that many New World wines go toe-to-toe with the best France has to offer!) are using grape vines that were originally imported from Europe.
In addition, there are several species of grapes that are native to North America, the best known of which is probably the Concord grape. Unfortunately, due to their flavor profiles these species don't tend to be used for winemaking (one exception being Manischewitz and other sweet ritual wines). However, they have a natural resistance to phylloxera, which saved vitis vinifera from decimation and likely extinction through the grafting technique.
Hey now. You send me a palatable French wine and I'll send you a palatable American one =)