Can weight loss be done with food only? Sure. Is it affordable or attainable for the average american? Not really.
Fresh veggies are the cheapest thing in a grocery store. And you don't need to buy the organic/exotic ones to "eat healthy" from the perspective of losing weight.
You know what's expensive? Junk food. For the price of a single bag of chips I can buy several pounds of carrots.
America has a food regulation problems that drive a variety of obesity-related problems. America also has a laziness problem that is almost mandated by suburban sprawl and automobile-oriented urban centers.
Most Americans have severely limited food choices. But even those who don't are subject to marketing and sugar-addiction that leads to bad food choices even when better choices are at hand.
Still, it's ok to hold accountable a person for buying and drinking two 20-ounce sugar-coffees for breakfast and then heading to the local strip mall to eat dinner at the restaurant that only serves 2000mg sodium meals--the affordability of vegetables was not the problem.
Keeping the same budget, you could most probably eat healthier in smaller quantities.
Although cost might be still be an issue for some, the main problem is education and culture, and these are harder to change: even if you can afford healthy food, you're unlikely to seek it if you have no appreciation for it.