The short answer is no. The long answer is no (with one or two exceptions).
Consider two diets: in the first one, you only eat green salad; in the second one, you only eat McDonald's hamburgers. Although I certainly wouldn't recommend the second diet, it is unquestionably healthier than the first. Protein (like fat, but unlike carbohydrate) is an essential nutrient, and green salad does not contain any. Although you might eventually catch scurvy through lack of vitamin C on the hamburger diet, you would be dead in a matter of weeks (or a couple of months at best) on the green salad diet. But how can this be? Green salad is "healthy", whereas McDonald's hamburgers are "unhealthy".
Before I explain why I don't think there's such a thing as an unhealthy food, it is necessary to define the term 'healthy'. A reasonable definition of 'healthy' (as applied to things humans do or eat) is 'conducive to the avoidance of morbidity'. In other words, one thing is healthier than another thing if it offers greater protection from infection and disease.
A category error is the assignment of a property to an object that could not meaningfully possess that property; the phrase was coined by the philosopher Gilbert Ryle. An example of a category error would be describing a carrot as intelligent, or––as in the case of Ryle's original example––conceptualising Oxford University as a single physical entity. I would argue that (in general) it is a category error to assign the property of being healthy to a particular food. I.e., it does not make sense to say, "green salad is healthy" or "hamburgers are unhealthy", just as it does not make sense to say "carrots are intelligent" or "Oxford University is a single physical entity". Comparatively, I would claim it does make sense to speak of diets as being healthy or unhealthy.
One reason why it is not meaningful to refer to a particular food as unhealthy is that, for any substance, there is always a quantity which, if ingested, would not cause any adverse consequences. For example, you would not suffer any adverse consequences if you happened to ingest 0.00000000000000000001g of plutonium. As a friend pointed out to me, this is exactly the point chemists have in mind when they say, "it's the dose that makes the poison." At the other end of the spectrum, there is always a quantity of a substance which, if ingested, would kill you. Trivially, even nutrients that are readily excreted by the body would build up and cause death if sufficiently large quantities (e.g., millions of kg) of them were ingested. More interestingly, there are rare cases of water overdoses.
A fair response to the preceding argument is that when a person describes a food as unhealthy she is usually referring to a typical portion size of that food. For example, someone might say, "it would be unhealthy for me to eat this piece of cake." Assuming it is possible to set bounds on what a typical portion size constitutes, I would argue that this response is unpersuasive.
The first reason why it is not persuasive is that, for any typical portion size (of any food), there is a frequency of ingestion of that portion size that would not lead to any adverse consequences. For example, suppose you think that it would be unhealthy to eat a piece of cake. Clearly, if you only did so once, the consequences for your health (as defined above) would be utterly negligble. The second reason why is that the effects of injesting portions of particular foods on health are not additive. In particular, the impact upon your health of eating a portion of some food depends on what else you've eaten recently. Suppose you believe that it would be unhealthy for you to eat a piece of cake. Would you avoid consuming a piece of cake on health grounds if you were starving in the desert and there was nothing else to eat?
At this point, a conceivable response is that when a person describes a food as unhealthy she is actually referring to a typical frequency of injestion of a typical portion size of that food in the context of a typical diet. In that case, I would agree, her claim is meaningful. But notice that she is now talking about diets and not individual foods. She is saying, "given what else I generally tend to eat, adding a certain number of portions of a particular food would be unhealthy relative to not doing so", which is just the same as saying "my usual diet plus a certain number of additional portions of food is less healthy than my usual diet as it stands." In other words––as stated above––it only makes sense to refer to diets as healthy or unhealthy, not individual foods.
Clearly, a diet made up entirely of McDonald's hamburgers (e.g., four quarter pounders per day) is much less healthy than a diet that includes fish, vegetables, grains, etc. However, it does not follow that as soon as you eat a McDonald's hamburger you somehow become unhealthy. There are really only two characteristics a diet needs to possess in order for it to be healthy. First, it should not be too low or too high in calories; eating 2500 calories per day is much healthier than eating 5000 calories per day (unless you're doing a great deal of excerise or you happen to be extremely tall). Second, it should be balanced. It is important to eat a variety of foods: vegetables, fruit, nuts, grains, and a range of meats; or if you're a vegetarian, a range of pulses and beans.
Before discussing one caveat to my argument, it is worth addressing what I believe is another source of confusion in the present context, namely the difference between 'unhealthy' and 'fattening'. People often refer to foods as unhealthy when what they really mean is fattening. Consider all the individuals in the developed world who are not close to being morbidly overweight. Adding a couple of portions of cake to those individuals' diets each week would not have any discernible impact on their health, but would make them a little bit fatter. In the case of fatness (as opposed to health), the effects of injesting portions of particular foods are more-or-less additive. Because of this fact, it is meaningful to describe one food as being more fattening than another. And what is meant by 'fattening' is something like 'moorish and unfilling'. For example, 500 calories of chocolate cake has very different effects on the neurological pathways governing satiety than 500 calories of boiled potatoes.
An important caveat is that there are some individuals who will suffer adverse conseqeunces from consuming even a single portion of a particular food. Such individuals may have allergies (e.g., to peanuts) or may have diseases such as diabetes, which prevent them from eating sugary foods when their blood-glucose level is high. Obviously, however, these people are quite capable of injesting very small quantities of the relevant foods.