Morton was born in Falkirk, Scotland. He began acting in his local children's theatre. After graduating from Mountview Academy of Theatre Arts in London, Morton worked for Profit Share Theatre and for film and television in the UK.
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 received a Whatsonstage and a Laurence Olivier Award nomination for the role of Boy George in the UK and also earned Tony Award, Drama Desk Award, Outer Critics Circle and Drama League Award nominations, as well as the Theatre World Award (for Outstanding Broadway Debut) in the US. He played the role of Ligniere in the play "Cyrano de Bergerac" from 12 October 2007 to January 6, 2008 on Broadway at the Richard Rodgers Theatre. Morton performed in the musical revue Sondheim on Sondheim, conceived and directed by James Lapine in 2010, which premiered on Broadway at Studio 54 and was presented by the Roundabout Theatre.
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 March 1, 2007 to May 6, 2007. He appeared in the musical Atomic which ran at the Acorn Theatre from July 13, 2014 to August 16, 2014, in the role of J. Robert Oppenheimer.
In regional theatre, he played the title role in The Who's Tommy at the Bay Street Theatre in Sag Harbor, New York in 2006. In 2010 he played Anatoly Sergievsky in the Signature Theatre (Arlington, Virginia) production of Chess. In 2011, he played the role of Launce in Two Gentlemen of Verona at Shakespeare Theatre Company. He appeared in the play Heart of Robin Hood, written by David Farr, which had engagements in Winnipeg and Toronto in December 2014 through March 2015, in the role of Prince John.
Morton won a Helen Hayes Award for his performance as Leo Frank in the musical "Parade" at Ford's Theatre in Washington, D.C. in September to October 2011.
Morton is also credited with performing as Renfield in the 2011 Studio Recording of Frank Wildhorn's "Dracula, the Musical", who sings "Master's Song" and "Master's Song (Reprise)".
As of November 29, 2016, he has assumed the role of Hedwig in Hedwig and The Angry Inch national tour. On July 2, 2017, Morton finished his last show of Hedwig tour.
On July 12, 2017, Euan's son Iain Armitage announced via Facebook video on his Facebook page Iain Armitage: Iain Loves Theater, that his father will be taking over the role of King George in Hamilton on Broadway sometime in the coming future.
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.
In March 2006, he released his debut solo CD, NewClear, for which he toured in the U.S. In May 2011, he released his second solo studio album Caledonia - The Homecoming.
He has been married to producer Lee Armitage, daughter of political figure Richard L. Armitage, since 2004. Their young son Iain is an internet theatre critic who has discussed Broadway shows on television talk shows and conducted Red Carpet interviews for SiriusXM radio before the 2015 Tony Awards. They maintain homes in Manhattan and Arlington, Virginia.