If a tax on fatty foods means that healthier foods are less costly than fatty foods, then yes, it will promote better public health. A better plan, however, would be to simply lower the cost of healthy foods to make them cost less than fatty foods currently cost to help promote better health.
Instead of creating better public health, a tax on fatty foods would ultimately create an underground economy dealing in fatty foods. 'Sin taxes' have created an underground economy for cigarettes in the United States. People in general find a way to gain access to the foods and substances that they desire.
Taxing fatty foods would not promote better public health. It would just make foods more expensive for people with lower incomes. Fatty cuts of meat tend to cost less than leaner cuts. Convenience foods are easier to obtain for people with limited funds and transportation. Making healthier foods more accessible for low income families would promote better public health.
The simple fact is that a person will only eat healthy if they are interested in eating healthy. Adding additional taxes may make some small differences at the margins of the statistics, but overall people will just pony up the extra cash for their hamburgers, pizza, and whatever other fast food they enjoy. A better strategy would be to offer healthy alternatives at the fast food places, so those who are health-concious will have options no matter where they are.