No, he didn't. He was too young to be drafted for World War 2, was in college during the beginning of the Korean War, and was already in his mid-30s and acting on children's shows when the Vietnam War went into full speed. There are rumors he was a sniper in Vietnam but they're about as true as the false rumors he was a child molester.
I looked this up once. I thought it would be a cool fun fact, but from what i found that is not true.
It also said that he did not have tattoo sleeves under his sweaters.