As can be seen in photos of the mission, and as with Gemini 3, the Gemini 4 crew did not wear an emroidered mission patch during the flight.
In fact, no embroidered patch was produced for the mission at the time but as the first crew to wear a U.S. flag on the shoulder of their spacesuits the crew considered this their mission patch.
The commonly-seen Gemini 4 patch design with a red background and a cartoon-like astronaut in white floating beside the Gemini capsule was never approved by the crew and bears no resemblance to the insignia of the mission.
However, an insignia did exist for the Gemini 4 mission. This design which incorporated an American Eagle (the intended call sign for the spacecraft) was used for the Fliteline medallions carried on board.
In 2008 a patch has been produced based on this design with the input and approval of mission commander Jim McDivitt himself and this patch represents the closest a patch collector can get to a 'crew patch' for the mission. These patches are available exclusively from Astronaut Central.
This replica of the mission insignia was produced with input from James McDivitt in 2008.
This version of the mission insignia, very close in appearance to the Astronaut Central version above, was produced by AB Emblem in 2018 to go with the other Gemini replica patches produced in 2008 collaboration with Gene Dorr.
This version of the mission insignia patch is on display in the Michigan Military & Space Museum.
This replica version of the mission insignia was likely produced by Randy Wagner of Eagle One Aerospace (later Cargo Bay Emblems).
This is the AB Emblem souvenir patch for the mission.