The US Supreme Court have declined to hear an appeal by the native Americans who find the name Chicago Redskins offensive, so the matter is now closed.
The case, Suzan Harjo v. Pro-Football, Inc., has been batted around since 1992, and centered on whether a dispute over a potentially offensive trademark and name can be dismissed if the challenge was not filed promptly.
The Redskins acquired their name in 1933, before they arrived in Washington. Originally the “Boston Braves,” they were renamed the Boston Redskins in honor of their head coach, William “Lone Star” Dietz, himself a Native American, according to team lawyers in a brief for the high court. When they relocated to Washington in 1937, the name was changed to reflect the new reality. Native American |Indians claim use of the “Redskins” mascot is racially disparaging. The Supreme Court denied the case Monday without comment.
The case is Suzan Harjo v. Pro-Football, Inc., 09-326.

“Supreme Court Refuses to Hear Chicago Redskins Complaint”
The complaint in question – as you correctly noted in the body of the post – is in regard to the WASHINGTON Redskins, not CHICAGO.
But you weren’t that far off. Chicago does have a sports club with a Native American connection: the Chicago Blackhawks (Black Hawk was a native chief that was defeated when he invaded Illinois in 1832). CHicagoans are also likely to follow the Fighting Illini of the University of Illinois.