At the end of the poem Sir Gawain and the Green Knight, what is the Green Knight's opinion of Gawain?
The short answer to your question is that the Green Knight approves of Sir Gawain or he would be dead. The challenge which begins the tale of Sir Gawain and the Green Knight is a simple one--Sir Gawain will allow someone to chop his head off today, but a year from now he must find the Green Knight's home and allow his own head to be chopped off by the Green Knight. Needless to say, there were very few takers for this challenge--only two, in fact. King Arthur and the young but passionate Sir Gawain.
All went as planned, and a year later Sir Gawain was in the home, though unbeknownst to him, of the Green Knight. Gawain did fulfill his obligations in an honorable way--almost. In the deal they made, Gawaindid withhold the scarf from his host, and he received a nick on his neck as his punishment. But that was it. Rather than chop Gawain's head off, as was his right, the Green Knight treated him like a man of honor. He said:
"Therefore I bid thee, knight, come...and make merry in thine house; ...I wish thee as well as any man on earth, by my faith, for thy true dealing."
The Green Knight admired him enough to invite him back to his home, though Gawain demurs, and wishes him blessings on his journey.
[S]o they embraced and kissed, and commended each other to the Prince of Paradise, and parted right there, on the cold ground.
This is the picture of courtly behavior between two knights. There is little doubt that Sir Gawain passed the test of honor given him by the Green Knight and he was treated accordingly--with respect and commendation.
The Green Knight ultimately approves of Sir Gawain and invites him to celebrate with a feast in his hall. Toward the end of the story, the Green Knight reveals that he knows about Gawain taking the magic girdle and wooing his wife: these were traps laid by the Green Knight. When he learns this truth, Gawain takes off the girdle and confesses to his "misdeed" (line 475). He feels remorseful and wants to win back the Green Knight's respect. The Knight says he forgives Gawain:
You are so fully confessed, your failings made known,
And bear the plain penance of the point of my blade,
I hold you as polished as a pearl, as pure and as bright
As you had lived free from fault since first you were born. (480-483)
The Knight describes Gawain as "pure"; he recognizes that Gawain only tries to wear the magic girdle to fight the Knight because it seems like he cannot win without some magic, and Gawain values his own life. Of course, there was trickery on both sides, so it would be a bit hypocritical for the Knight to hold Gawain's actions against him. Gawain has confessed his wrongs, and the Knight now sees him as faultless. He even gives Gawain the girdle as a prize and then invites him to feast with his men. Ultimately, the relationship between Gawain and the Green Knight ends on a positive note.