No, Gawain is often portrayed as a foil for the other "true" knights. Thus, he does not fit the mold, but by presenting himself as the opposite, he defines the medieval hero via these attributes.
For instance, when asked to choose a girl for a guide in the section titled: "The Tale of King Arthur", he chooses the youngest and most beautiful. He makes it clear that his intentions are sexual in nature and manipulative, and he is shocked when she chooses the dwarf over him in the forest. He then later refuses to help Sir Pelleas, who is being badly defeated, and even though his lady pushes him to it, he refuses her wishes as well. Eventually, his "solution" is to sleep with Sir Pelleas' love in the castle as a means by which to solve his "sorrowful situation". Now, this is just a paraphrased set of events in the beginning of Morte Darthur, but Sir Gawain has already created a character which negates the principles of Arthur's court.
He refuses a lady's wishes, desires his honor over others regardless of the necessary means, is not chaste, shames himself publicly, is prideful, and lies. Meanwhile, Marhault and Ywain put him to shame in this section. Gawain may be presented somewhat positively in other renditions (such as in "Sir Gawain and the Green Knight") but not in Morte Darthur. Both Gawain and Kay are fervently used as massive chivalric foils.