Do natural foods contain all organic ingredients?

I can answer your question in one word.  No.  Natural only means the manufacturer started with a natural source and that the product is not artificial.  For example, natural applesauce may contain only apples and apple juice which are both natural products.  According to Jesse Ziff Cool’s book, Your Organic Kitchen, Apples, on average are treated with 32 different pesticides.  Furthermore, products labeled natural may contain GMOs (genetically modified organisms).  Natural foods that do not comply with organic standards can still be labeled with truthful information such as free-range and hormone-free.
Organic on the other hand, is a highly regulated term.  Under the U.S. Department of Agriculture guidelines for organic food Organic meat, poultry, eggs, and dairy products come from animals that are given no antibiotics or growth hormones.  Organic food is produced without using many pesticides that are used in conventional farming and fertilized without petroleum and sewage sludge-based fertilizers. Products labeled organic must be grown at a farm that has been inspected to make sure the farm is adhering to USDA organic standards.