This would be a natural sentence only if you meant if someone gave a person anything to eat or if someone got his dinner from any source, for example- school canteen, restaurant etc.
However, if you wanted to ask if someone already ate his/her dinner/lunch/breakfast or not, then you would have to say "Did you HAVE your dinner/lunch/breakfast?" or "Did you EAT your dinner/lunch/breakfast?"
Hope it was easy to understand, right? Ask me any question you want, don't hesitate! ^_^