Of course, that still leaves many unanswered questions. What do we mean by environment? Just C02, or is it water quality, or deforestation, or what? Is the label voluntary or mandatory? How will food companies respond to the label? What do consumers understand about the label? And so on.
In principle, it is possible to imagine something like a nutrition facts panel for environmental issues. However, the two are not as analogous as might first appear. First, scientists have a pretty good idea how to measure the fat, carb, and protein contents of food, whereas measuring C02 or deforestation impacts is tricky business with a lot of uncertainty. Moreover, the nutritional content of a processed food is relatively stable regardless of where the raw ingredients came from, which plant or facility was used to manufacture it, how it got to the store, or how you transported and cooked it. None of this would be true for an environmental label, which would require more more extensive (and more costly) monitoring and tracing, and if it is at all accurate, one could have two Wheaties boxes that are nutritionally equivalent but with very different environmental impacts. That may be all the more reason to inform consumers, but the point I'm trying to emphasize here is the much higher cost and greater uncertainty in informing about nutrition vs the environment.
Finally, an perhaps most importantly, nutritional outcomes are, by and large, what we economists would call "private goods." The nutrient contents of the foods I eat affect me personally and not others (let's put aside Medicare/Medicare, which is another issue I've touched on here, here, here, and here). In these cases, the effects of a label on my choices, and ultimate welfare consequences are more straightforward. Let's compare that to environmental labels, which signal attributes associated with public goods and possible externalities, where we suspect there are likely to be problems with coordination, free riding, etc. I suspect most economists would tend to look toward getting the property rights or the prices right as the "solutions" in these cases rather than looking toward labels (here's a paper I wrote on that issue).
Finally, I'll note there is a long literature in agricultural economics on food labels - focusing on when and under what conditions labels enhance social welfare. The results of this literature are hard to summarize (meaning the effects are complex). Here are a few good places to start if you're interested in the topic.