To understand the similarities between Beowulf and Sir Gawain and the Green Knight, it is important to understand what epic poetry is.
An epic poem is a long narrative poem; it usually focus on a story that reflect heroic events and deeds...Oral poetry may qualify as an epic, and Albert Lord and Milman Parry have argued that classical epics were fundamentally an oral poetic form.
Oral poetry is a tradition of passing stories and histories by word-of-mouth, before stories were written down. This was the job of one member of a community, man or woman. The job was passed down from parent to child, and this person, often called a scop or bard, recorded important events, such as battles, and stories, etc. Beowulf was passed down in the oral tradition long before it was written down.
The first epics are known as primary, or original, epics. One such epic is the Old English story Beowulf.
Epic: a long narrative poem in elevated stature presenting characters of high position in adventures forming an organic whole through their relation to a central heroic figure and through their development of episodes important to the history of a nation or race.
With these descriptions in mind, look at specific pieces of literature. I believe there is information to support that both Beowulf and Sir Gawain and the Greek Knight are epic poems.
Both Beowulf and Gawain are "central heroic figures." The stories also have several parts or episodes.
In Beowulf, there is the battle with Grendel, Grendel's dam (mother) and the dragon at the end. The story of Gawain has episodes as well. First Gawain faces the Green Knight during the Christmas celebration at King Arthur's court. Next Gawain travels to the chapel to meet the Green Knight a second time. Arriving early, he is invited to spend the holidays with Bertilak and his wife. Finally, Gawain goes to meet the Green Knight at the Green Chapel.
Both poems focus on "heroic events and deeds." Beowulf faces three adversaries. Upon defeating the first two, he brings peace to the land of the Danes. Years later, after a dragon is disturbed from his sleep and begins to prey on the surrounding land and its inhabitants, Beowulf, poisoned as he is, kills the dragon (with the help of Wiglaf), saving his people from the scourge of the "fire-drake."
Gawain performs heroic deeds. He first faces the Green Knight, believing he will kill him. When he fails, he gives his word to meet the Green Knight a year later to face his part in the challenge. Honor-bound, Gawain does so. He proves his honor by avoiding Bertilak's wife's advances; he returns all he receives to Bertilak (but the magic belt, for which Bertilak forgives him), and stands fast as the Green Knight brings the ax down three times toward his neck—because he was honor-bound to do so.
Although written in different time periods, the two poems, Sir Gawain and the Green Knight and Beowulf [share] many similarities. Both of these poems share the same common ideas, such as the type of qualities that heroes possess. Each of the two poems also has a main character that exhibits these qualities of bravery, honor and truth.
To be able to defend the premise that Beowulf and Gawain are similar characters and that their stories are both told in the format of an epic poem, one need only compare the stories, and the common characteristics of epic poetry they share.
Additional source: http://www.enotes.com/sir-gawain-and-the-green-knight/summary