Morton appeared in The Silent Treatment at the Finborough Theatre, London, in 2001. Immediately following, he played the role of Boy George in the West End musical Taboo in 2002, and also played the role on Broadway in 2003.

He has been married to Producer Lee Armitage, daughter of political figure Richard L. Armitage, since 2004. Their son Iain got his start as an internet theatre critic. He discussed Broadway shows on television talk shows and conducted Red Carpet interviews for SiriusXM radio before the 2015 Tony Awards. Iain is currently playing the title character in the television show Young Sheldon. Morton and his family maintain homes in Manhattan and Arlington, Virginia.

In March 2006, he released his debut solo CD, NewClear, for which he toured in the US in May 2011, he released his second solo studio album Caledonia – The Homecoming.

Off-Broadway, he appeared in the title role of Tony Kushner's adaptation of Brundibar at The New Victory Theater in 2006. He won the 2006 Obie Award for his appearance in Measure For Pleasure at The Public Theater. He played the title role in the musical Caligula: An Ancient Glam Epic at the inaugural New York Musical Theatre Festival in September 2004. He appeared, along with Alfred Molina, in the Roundabout Theatre Company's new production of Howard Katz by Patrick Marber, which ran Off-Broadway at the Laura Pels Theater from 1 March 2007 to 6 May 2007. He appeared in the musical Atomic which ran at the Acorn Theatre from 13 July 2014 to 16 August 2014, in the role of J. Robert Oppenheimer.

Morton has performed numerous audio books, including among others, Fool (2009), Sacré Bleu (2012), and The Serpent of Venice (2014), all written by Christopher Moore, in addition to the Kilo Five trilogy by Karen Traviss and Carry On by Rainbow Rowell (2015). He also did voice work in the 2009 animated movie My Dog Tulip, and voices the male Sith Inquisitor in the MMORPG Star Wars: The Old Republic, created by BioWare and LucasArts.

Morton is also credited with performing as Renfield in the 2011 Studio Recording of Frank Wildhorn's Dracula, the Musical, singing "Master's Song" and "Master's Song (Reprise)".

Morton assayed the role of Sherlock Holmes in Ken Ludwig's Baskerville: A Sherlock Holmes Mystery at the Old Globe Theatre in San Diego, California in 2015.

As of 29 November 2016, he has assumed the role of Hedwig in Hedwig and The Angry Inch national tour. On 2 July 2017, Morton finished his last show of the Hedwig tour.

On 12 July 2017, Euan's son Iain Armitage announced via Facebook video on his Facebook page Iain Armitage: Iain Loves Theater that his father would be taking over the role of King George in Hamilton on Broadway sometime in the coming Future.