The short answer is "yes," but it is a qualified "yes". Diet affects the health of the skin in general. The skin is an organ of the human body just like the heart, the lungs, the liver, the kidney, etc. And the skin, like other body organs, requires certain nutrients to function correctly. You probably know that foods that whole grain foods are considered good for promoting heart health, for example.
There are many nutrients that are important for promoting healthy skin. Just a few of them are Vitamin B6, chromium, selenium, vitamin A, zinc, and EFAs (Essential Fatty Acids).
Because the usual Western diet contains few of these nutrients, I can say that diet does have a great deal of effect on acne. Americans and particularly American teenagers, eat a diet that is high is saturated fats, high in sugar, high in carbohydrates, and low in protein, and many of the vitamins, minerals, and trace elements that are needed for a healthy body, including a healthy skin.
To be fair, it is also important that I tell you that there is no food that is known to cause acne. There is no substance that CAUSES acne. Acne is caused by the sebaceous glands producing an overabundance of sebum and the sebum attaching itself to dead skin cells. Food doesn't cause acne. But the fact is that the LACK of the nutrients needed to promote and maintain a healthy skin certainly can and does contribute to the problem of acne.
If you want healthy, glowing, acne-free skin, you need to eat a diet that is high in fresh fruit, fresh vegetables, whole grains, fish, and shellfish. This diet doesn't guarantee that you will be acne free, but it does mean that your diet won't be contributing to your acne problem.