This same rumor has often been applied to boyish country singer-songwriter John Denver (among others), and it's just as false when told of Fred Rogers. Not only did Fred Rogers never serve in the military, there are no gaps in his career when he could conceivably have served in the military — he went straight into college after high school, he moved directly into TV work after graduating college, and his breaks from television work were devoted to attending the Pittsburgh Theological Seminary (he was ordained as a Presbyterian minister in 1963) and the University of Pittsburgh's Graduate School of Child Development. Moreover, Fred Rogers was born in 1928 and was therefore too old to have enlisted in the armed services by the time of America's military involvement in Vietnam.
Fred Rogers always wore long-sleeved shirts and sweaters on his show to conceal the tattoos on his arms he obtained while serving in the military.
As noted above, Fred Rogers never served in the military, and he bore no tattoos on his arms (or any other part of his body). He wore long-sleeved shirts and sweaters on his show to maintain an air of formality — although he was friendly with the children in his viewing audience and talked to them on their own level, he was most definitely an authority figure on a par with parents and teachers (he was Mister Rogers to them, after all, not Fred), and his choice of dress was intended to establish and foster that relationship.