The 23-year-old, who can also play in defence, was not offered a fresh deal by the Whites this summer.
White made 85 league appearances for United but only featured once in 2014-15 because of an ankle injury.
He told the Millers website: I was at Leeds for a long time but coming here was a great opportunity to get back to playing football and I'm delighted to be here and I can't wait to get going.