According to a KGN interview with Randy Pitchford, Gearbox were approached by Activision to develop a new Call of Duty but they turned down the opportunity because Pitchford saw "no real challenge" in making "what is expected of a Call of Duty".
CVG spotted the KGN interview in which Pitchford discusses the motivation behind the projects Gearbox works on:
[Meine Deutche ist nicht gut is all quotes have been taken from Google Translate's read of the German interview.]
"For me, there are two scenarios in which a project makes sense for Gearbox", said Pitchford. "First, if there would have been the game without us ever. Or two, when we can contribute to an existing brand something new, a kind of unique perspective and a new approach. But that was not the case [with Call of Duty]. I think about the people delivering the "Call of Duty", which they want, you have to play by the rules of the series. You have to do what is expected of a "Call of Duty". In this I see but no real challenge for us. It would strengthen our reputation as a studio no further, it would not be really motivating for our team."
While the wording isn't ideal, we can pick out what exactly Pitchford was getting at. Gearbox are interested in either developing a game which is intrinsically of their style, a game that no other developer could make quite the same - Borderlands for instance, with it's eclectic characters, hodge podge world, and weapon fixation, couldn't have been developed by anyone else. Or, they want to add to an existing franchise, again, in a way unique to them. (We'll need to wait till later this year to see how their take on the Alien franchise differs from Rebellion and Monolith's later this year.) So designing a Call of Duty game which Activision will be overseeing with an exceptionally watchful gaze (it is one of their main money makers and any deviation that could lose them money would unlikely be approved) simply didn't appeal to Pitchford and his studio.
This is slightly frustrating because seeing what another studio could bring to the franchise that always appears slightly stale would be interesting. Though, as Pitchford states, any studio, no matter how competent, would be unlikely to receive a free rein.