Only those who were citizens from countries at war with the United States during the First World War registered. Your grandfather, being a U.S. citizen, would not have been required to register. However, depending on his age, he might have registered for the WWI draft. For information on finding his draft record see...
How to Find World War One Draft Cards 1917-1918http://www.genealogybranches.com/worldwaronedraftcards.html
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