What makes King Arthur a hero in Sir Gawain and the Green Knight?
Let us remember that this poem doesn't actually focus on King Arthur as the central protagonist. Therefore King Arthur is presented as something of a minor character given the focus that is placed on Sir Gawain and the eponymous Green Knight. However, the introduction does clearly state various reasons why we might regard King Arthur as a "heroic" figure. The narrator describes Arthur as the most "courteous" King and also credits him with bringing peace and order to Ancient Britain and establishing Camelot which is described as a city of wonders. In addition, we could argue that Arthur, by insisting he is the last to be served at banquets, shows great humility in his kingship.
Of course, Arthur's heroic nature is shown in the way that he is the first to respond to the Green Knight's challenge. When his knights stay silent after the words that are uttered by this mysterious personage, it is Arthur who seeks to reclaim the honour of Camelot and of the Round Table by responding to the Green Knight's threats:
Give me now this gisarm, by God's sake, and I will grant thy boon that thou hast bidden.
Arthur shows himself therefore to be heroic both in character and in deed and is therefore rightfully refered to as the most "courteous" of kings and secures his heroic status.