Celtic announced the passing of Miller in a statement on social media.
Miller made a name for himself in Australia, playing for A-League sides Perth Glory, Brisbane Roar, Melbourne City, while also spending time earlier in his career with Manchester United.
He won 21 caps for the Republic of Ireland over 16 years, breaking through at Celtic before he moved to Old Trafford in 2004. He played in a total of 76 games with Hibernian between 2009 and 2011.
It was reported in November that Miller was suffering with cancer, his former clubs rallying around him and his family.
Miller's death brought tributes from a number of his former clubs – including Leeds United, Hibernian and Sunderland as well as Roar and City.
Miller played a total of 68 A-League games, 49 with Perth Glory between 2011 and 2013.
He won the A-League championship with Brisbane Roar in 2014, playing 74 minutes in a 2-1 win over Western Sydney Wanderers.