Would you consider Sir Gawain a hero in his pursuit of the Green Knight?
I think Sir Gawain could be looked at as a hero because he travels to meet the Green Knight's challenge knowing that he faces certain death.
In searching for the characteristics of a hero of the fourteen century, the time at which the tale of Gawain and the Green Knight emerged, we find he is a folk hero. This kind of a hero can be someone who actually lived or be mythological in nature.
The single salient characteristic which makes a character a folk hero is the imprinting of the name, personality and deeds of the character in the popular consciousness.
The story of Gawain in this tale is separate from his presence in other Arthurian legends, and in this way, he has become familiar to the "popular consciousness." Because his life is not based upon actual historical occurrences, Gawain's "characteristics and deeds" have been "exaggerated to mythic proportions." The characteristics of Gawain are those associated with King Arthur and his knights of the Round Table. While Arthur may have lived in a setting much older than the one Thomas Mallory places him in, in Morte de Arthur—with warriors and chieftains as companions rather than knights—there was no Round Table. This was a medieval ideal, where no one person sat above the others. Arthur and his men would have been of a much less "civilized" age, though Arthur easily may have been a hero in his day. We have no way to know that his knights as named ever existed, and can assume that Merlin was fictious as well. As King Arthur is considered apocryphal, so too then is Gawain. This means they are "of questionable authenticity" (though there are some early historical references to Arthur—he may actually have lived). The characteristics of the Arthurian legends have been passed on to Gawain, and are present in this tale, when Gawain faces the Green Knight.
Chivalry pays an enormous role in the behavior of Arthur and all his knights. This concept is specifically medieval. These men went through "individual training" and were intent on providing "service to others." (This would include a damsel in distress, etc.) Honor was another major element of the chivalric code. In the case of Gawain and his initial challenge of the Green Knight in Arthur's court one Christmas, he is honor-bound to meet the Green Knight one year later to submit to a "challenge" from the seemingly supernatural knight whose head Gawain has attempted to cut off.
The Arthurian heroes are men with great bravery, as well as human frailties. However, this is something that perhaps make them more heroic and endearing as we read about them. Gawain knows he must keep his word as a member of Arthur's court. He refuses to give into the seduction of Bertilak's wife, but he is also human enough to fear death, and so while he hides the magical belt that Bertilak's wife gave him, Gawain maintains his code of honor when found out by staying still while the Green Knight takes three swings at his neck.
Gawain is not perfect, but heroes generally are not. However, he does his best, showing bravery, integrity and a strong moral code, to live an exemplary life that brings honor to Arthur and the knights of the Round Table.
Yes, I agree with you that Sir Gawain, King Arthur's nephew and a knight of the Arthurian Round Table, may be considered a heroic adventurer in his pursuit of the mysterious Green Knight. The way Sir Gawain accepts the challenge of the Green Knight's beheading game, and then undertakes the perilous adventure to receive the blow back at the Green Chapel, and the way he exchanges courtesies with the Lord and Lady Bertilak, suggest that the knight represents the ideals of chivalry, courage and heroism. First, he accepts the challenge to behead the Green Knight. Second, he keeps his promise to visit the Green Knight's chapel in order to receive back. Third, he behaves most courteously with the Lady who attempts to seduce the Christian knight. Although Sir Gawain's decision to keep back the gift of the girdle for the sake of personal security is looked upon as an act of shame, he returns to Camelot victorious after completing his mission of pursuing the Green Knight.