To add to my colleague's assessment of Gawain as an example of the "ideal Christian knight," I would suggest that while Sir Gawain may see himself as a failure, he is actually considered a success in this regard by everyone else concerned.
It is true that Sir Gawain stumbled when he took the green girdle and did not present it to Bertilak as was promised in their pact, but the sin seems minor in the eyes of even Bertilak himself, who, after Gawain's show of shame, proclaims him completely absolved of his minor shortcoming and invites him back to his home to celebrate the New Year.
In fact it is ONLY Sir Gawain who sees his failure to give the girdle to as a sign of his unworthiness. When he returns home, wearing the girdle as a baldric to both hide his scarred neck and serve as an outward emblem of his shame, the ladies of the court are entirely impressed with his bravery, and his fellow knights hold him up as a standard of bravery and knightly behavior. They even begin wearing bright green bands in his honor.
I would argue that Gawain is among the most qualified to the title of "Christian Knight." He bravely faces the Green Knight, follows through with his promise to take his strike, prays for guidance, resists nearly all temptation, and fully repents when confronted with his own sin. The fact that the only reason he faltered was to save his life does not make others judge him, but rather gives them reason to empathize with him. This small failure on Sir Gawain's part does not make him less of a Christian; it makes him more of a human.
The ideal Christian knight as defined by Sir Gawain and the Green Knight would have to live up to the code of chivalry. Chivalry would look poorly on those who did not have self-discipline when it came to lust or physical pleasure. It held in high value physical courage, loyalty to one's lord or king, and piety toward God and Christianity. Christianity values love and marriage. It is important as a knight that he be loyal first and foremost to God and his religion and just as importantly to his king.
Sir Gawain from the very beginning shows his loyalty to his king by taking the Green Knight's challenge in the name of King Arthur. Sir Gawain is essentially sacrificing himself by delivering a blow to the man in green knowing in a year and a day, he will also receive a blow with this knight's axe. If Sir Gawain had not taken this pact, the honor of King Arthur and his kingdom would be in question as the Green Knight mocks them.
On his journey, however, Sir Gawain does not live up to the integrity and honesty portion of being a good knight. He makes another pact with a lord whose home he stays at. The Lord, Bertilak promises him his day's returns hunting for whatever Gawain receives during his days at their home. After three days, Gawain receives kisses from Bertilak's wife each day. Each day, Gawain gives Bertilak the kisses. On the third day, in addition to kisses, the wife gives Gawain a sash which Gawain fails to deliver to Bertilak. Throughout his stay, Gawain refuses to be seduced by Bertilak's wife which is honorable and part of being a good Christian Knight.
In the end, Bertilak reveals himself as the Green Knight and delivers one, non-fatal blow, which is meant to punish Gawain for not revealing the sash to Bertilak. However, it is not fatal, because Gawain proved himself to be honorable by not giving into Lady Bertilak.
At the end, Gawain wears the sash on his arm as a symbol of his dishonest behavior and as a reminder to be penitent for this poor behavior. This is a sign of Christian piety - to be sorry for one's sins and to make up for sinful behavior.