Sosa is wanted in the United States for allegedly making false immigration statements by not admitting his military past in his home country of Guatemala.
He has both Canadian and American citizenship.
Justice Brian O’Ferrall ruled on Wednesday that Sosa's appeal is "hopeless" and there is no injustice in requiring him to answer the perjury charges in the United States.
Sosa is also wanted in Guatemala, where he has been accused of taking part in a village massacre in 1982 in which 251 men, women and children were killed.
Sosa was arrested in Lethbridge, Alta., in January 2011 while visiting family.