Rather than believing himself to be worthy or the best qualified to accept the Green Knight's game, Gawain is the one to take the Green Knight up on his challenge simply because he wishes to spare King Arthur from having to do it himself. In fact, the text supports the idea that Gawain believes himself to be lesser qualified than many of the knights in attendance; his is simply more ready than they are to protect the king from what would have been seen as certain demise.
Upon the Green Knight's entrance into the hall during the feast and explanation of his reason for coming, the onlookers, including Arthur's knights, are stunned into silence. Indeed, it is the silence that causes Arthur to be the one to accept the Green Knight to protect his personal reputation as well as the reputation of the court. It is only once Arthur accepts that Gawain steps in and states,
I am the weakest, I know, and of wit feeblest. / Least worth the loss of my life, who’d learn the truth. / Only inasmuch as you are my uncle, am I praised.
Thus, the humble and loyal Gawain is the one to trade blows with the Green Knight because of his willingness, not his worth.