Icardi infuriated sections of the Curva Nord ultras after a passage in his new autobiography suggested he had made himself a "hero" by standing up to one of the group's leaders.
The 23-year-old claimed he had argued with the fan after Inter's 3-1 loss to Sassuolo in 2014-2015, after his shirt - which he attempted to give to a member of the crowd - was thrown back at him.
Banners were displayed during Inter's 2-1 loss to Cagliari on Monday (AEDT) claiming Icardi's career with the club was "finished", while sections of the supporters cheered when he missed a penalty.
A club statement released on Tuesday confirmed Icardi had been sanctioned, although he is expected to retain possession of the armband, prompting the Curva Nord to claim the striker is not worthy of leading the team.
"For us as for everyone else - we hope - the captain of Internazionale should be a symbol," a statement read.
"Experience, charisma, humility, integrity, intelligence.
"[Armando] Picchi - [Giacinto] Facchetti - [Sandro] Mazzola - [Giuseppe] Bergomi - [Javier] Zanetti... basically the opposite of Icardi.
"We therefore cannot find agreement with the statement of the Inter hierarchy. We understand it, the logic, the motivation. But we can't accept it.
"It's not for us to decide though, to each his proper role. That's what we've always asked for, that's how it should be.
"We believe though that a club like Inter, outside of our views, shouldn't allow an individual like Icardi to wear the captain's armband.
"Icardi is a closed chapter, we don't want to talk about this anymore. HE IS NOT OUR CAPTAIN. HE IS NOT NOW, HE NEVER WILL BE."