The correct answer is loyalty to his king.
As you can see in the excerpt, Sir Gawain is having a conversation with his king about having a fight with the Green Knight in order to defend his honor. He never doubts his allegiance to the king - quite the contrary, he immediately offers his services as a way to prove how loyal he is to his lord.
Unfortunately, Sir Gawain somewhat failed in his quest, having been seduced by the Green Knight's wife. Luckily for him, he makes his amends and the King forgives him.