How does Sir Gawain fulfill the role of the Romantic Hero in Sir Gawain and the Green Knight?
Romantic heroes are full of positive, valiant attributes of bravery, loyalty, humility, courage, honesty, virtue, and valiane. Gawain possesses all of these traits. He is loyal to the king as he steps in to take defend Arthur and his kingdom against the insults of the Green Knight, he is humbled at the end at the knight's chastisement for slipping up on his promise, he is brave and courageous to accept the challenge given to the court, and on his various trials along the journey, he is honest (for the most part) in his dealings with the Green Knight, and virtuous with the lady of the castle, in rejecting her repeated attempts to seduce him. So, he possesses heroic attributes.
Romantic heroes also often fulfill a quest, challenge or adventure of some kind--they set off to rescue damsels in distress, to rid the kingdom of evil, or to combat mythical creatures that are causing mayhem. Also, they often have a lady or woman that they pay tribute to, meaning, they dedicate their powerful quest to that lady, in order to win her affection and love. The rules of courtly love insist that this woman is unattainable (meaning, married) and gives this gallant knight a token of her affection (a scarf, a glove, etc.) that the knight then wears on his armor as he goes into battle. Note how Gawain kow-tows to the queen before he takes off, and how he takes tokens from the lady of the castle into battle. So, Gawain is a romantic hero in the sense that he is following the codes and protocol of romantic courtly love also.
Gawain is handsome, strong, gallant, and a perfect gentlemen in his dealings with the ladies. He does not hesitate to defend his king and kingdom, and shows weakness only in the potential killing of another man, and in keeping a love token from the lady of the castle, all of which the Green Knight forgive him. I hope that these thoughts helped; good luck!