The 23-year-old has signed a three-year deal after the Millers agreed an undisclosed fee with the Scottish club.
His first bow in red and white could come in the weekend friendly at home to Leicester City.
Lindsay, who started his career at Celtic, was a key member of Ross County’s Championship-winning side last season. He made 48 appearances in all competitions and scored nine goals.
Paul Warne’s side will kick-off their Football League Trophy campaign earlier than most clubs this term after it was revealed the home clash with Manchester United Under-21s will take place on Tuesday August 6.
The New York Stadium encounter will kick-off at 7pm. Completing the group are Doncaster Rovers and Lincoln City.
Most opening group games are expected to be staged in the week beginning September 2 but international commitments for the Old Trafford youngsters saw the Millers clash brought forward to the first Tuesday of the campaign.