Sayfildin Tahir Sharif — who also goes by Faruq Khalil Muhammad Isa — is accused of murder and supporting a terrorist group that took part in suicide bombings in his native Iraq.
The Americans allege Sharif worked from Edmonton to help a Tunisian man enter Iraq in 2009 and detonate a truck filled with explosives at a military checkpoint, killing five U.S. soldiers.
His lawyers had argued that RCMP didn't allow Sharif access to a lawyer or interpreter the day of his arrest, and transcripts of police interviews show he didn't understand what was going on.
An extradition order was issued in June 2013.
As usual, the Supreme Court did not give reasons for its decision not to hear the case.