The 30-year-old completed the full 90 minutes of the encounter at the Stade Matmut-Atlantique, but complained about pain in his thigh after the match.
The former Newcastle defender underwent a series of fitness tests, with Bordeaux posting on Twitter: "Bad news for Mathieu Debuchy, who has suffered a thigh injury and will be out of action for several weeks".
Debuchy lost his place in the Gunners' first-team following the emergence of Hector Bellerin on the right of defence and has spent much of this season on loan in Ligue 1.
And Bordeaux head coach Rame later confirmed the defender will not feature at Euro 2016 for the host nation.
"Unfortunately, Mathieu will not play at Euro 2016," Rame stated at a news conference.
Debuchy moved to Bordeaux until the end of the season in the January transfer window in order to boost his chances of making France's squad for the upcoming finals on home soil and has since become a key figure at the southwestern side.
A member of France's Euro 2012 and 2014 World Cup squads, Debuchy last represented Les Bleus in the 2-1 friendly win over Serbia in September 2015.