Most physicians do not recommend putting your ASD child on a special diet as there is no diet which is clinically proven to cure Autism.
Restricting your child’s diet may worsen the nutritional status of your child as they already show food preferences that leave very little room for variety in eating. 

Gluten and Casein Hypersensitivity – A few autistic children can show hypersensitivity to gluten (found in wheat, rye and barley) and casein (found in cow, goats and sheep milk) which can aggravate their behavior.
A qualified dietician can identify the signs of hypersensitivity and prescribe a gluten free or casein free diet.

Vitamin and mineral deficiencies.
Optimum levels of vitamins and minerals are essential for normal functioning of the brain and its development.  Low levels of Vit B1, B6, B12 and Vit D are often reported among children on the ASD spectrum. Vitamin supplements are known to ameliorate the severity of ASD symptoms in some children.

However, this decision is best taken by the doctor who can evaluate your child’s health status.  The diet should be individualized to address the nutritional deficiencies if any or the food allergies present in the autistic child.