I have to admit that I really don't know that much about the Gemini Guidance Computer, except that it was designed and built by IBM. Wikipedia indicates that it was designed using the same RTL technology as that used in the Apollo Guidance Computer (AGC), but I have not seen any other definitive proof of this. It would be really interesting to get more info about this.
What I do know is that IBM was very much against using integrated circuits when MIT first proposed using this for the AGC. There are many documents where IBM attacked this decision, claiming that it was too unreliable. IBM actually proposed a different design using transistors and there is another document where MIT responded, showing that the IBM design was far less powerful than what was needed to land on the moon. IBM created enough doubt with NASA that they got the contract to develop the Apollo Abort Guidance System (AGS), which was a much simpler (and slower) transistor based computer of a similar size and weight. By that time, MIT already had two years experience designing with integrated circuits and the AGC3 was already functional. IBM was so insistent during the 1960's, on only using transistors in their designs that their Mainframe computers only started using TTL components in 1972.
The Gemini Guidance Computer was definitely used in flight before the AGC, but production AGC computers were available and used in training at least 18 months before the Gemini computer. The guidance computer used in the Minuteman II missiles was actually introduced at roughly the same time as the Gemini guidance computer, but also trailed the AGC development by 12-18 months.
The only reason why the AGC flew after the Gemini and Minuteman guidance computers were because of delays on other parts of the Apollo project. The AGC was definitely ahead of the pack by at least a year even though it was an order of a magnitude more complex and much faster. It was also much closer to a general microcomputer than the other special-purpose designs. It was used in 1972 for the first fully electronic fly-by-wire test onboard a modified F8 Crusader. It has also been used in US Navy's Deep Submergence Rescue Vehicle for navigation tests. It influenced the design of many other notable computers e.g. the PDP-11 and later general purpose microcomputers. You wouldn't be too far off, comparing it with a modern embedded microcontroller like the PIC. In my opinion, it is much closer to a general purpose microcomputer than any other computer built during the 1960's.
The title of the post is a bit ambiguous, so to clear it up. There is no doubt that the AGC was the first computer built using integrated circuits. It was also a general purpose computer that could do much more than just navigation. It was not the first computer in space, but the first computer built with microchips (that also happened) to fly in space.