Has he had anything to eat that could be causing the vomiting and diarrhea? If not, it could very well be viral gastroenteritis AKA a stomach bug.
There is no specific medical treatment for viral gastroenteritis. Antibiotics are not helpful for infections caused by a virus. However, there are a number of things you can do to be more comfortable and avoid dehydration.
Fluids—It is important to drink fluids to replace those you’ve lost when sick. Take small sips of water, suck on ice chips, or drink clear soda or noncaffeinated sports drinks.
Diet—Gradually begin to eat bland foods, such as toast, crackers, bananas, rice, chicken, and potatoes. Avoid dairy products, caffeine, fatty foods, and spicy foods.
Rest—Make sure you get enough rest while you’re sick and when you’re recovering.
If he doesn't seem to be improving, use your better judgment in regards to taking him to his doctor or if you see signs of improvement then you can wait a few more days--while keeping him on foods that will solidify his stool and drinks to keep him hydrated.
Avoid dehydration as much as possible because vomiting and diarrhea are both quick ways of losing fluids and electrolytes.