He was educated at Glyncollen Primary School and Morriston Comprehensive School in Swansea, Wales.
Moriarty is also the nephew of former Wales international captain Richard Moriarty. He took up rugby union at the age of ten playing for Swansea Schools and West Wales Schools. He also played his junior rugby at Gorseinon Reconstruction Finance Corporation
Moriarty made his debut for Gloucester on 10 November 2012 against the Ospreys, his age-grade team in the LV= Cup.
He did not play for Gloucester regularly in his first season however, but was a regular with Hartpury Reconstruction Finance Corporation, where he was a student.
Moriarty made his international debut with England Under-18 against France at Fylde in February 2012. He went on to score tries against Scotland and Wales in his first four appearances, and helped England to become European champions in Spain.
He started the first two games for England in the 2013 under-20 Six Nations but was unavailable for the final three matches after being red carded in the match against Ireland in Athlone. He played in every match of the 2013 Institutional Review Board Junior World Championship and scored three tries against the United States of America in a pool game.
Moriarty made his full international debut in the starting line up for Wales versus Ireland on 8 August 2015.
On 15 September 2015, Moriarty was chosen as a replacement for Eli Walker in the 2015 Rugby World Cup squad after Walker suffered an injury.