Why is it important to Beowulf and to his image as an epic hero that he face Grendel without a weapon?
Beowulf is a hero of epic proportions. He has already established himself through other heroic deeds. One example is his fight with a sea monster during a sea race. He possesses superhuman strength; he is the leader of a band of warriors; and he is on a mission to prove that he is worthy to be the King of the land of his birth. Additionally, the text tells us that Beowulf is good and "of God" while Grendel is evil; Beowulf believes he is so superior that it would be an unfair fight to fight with a weapon. While all this is true, Beowulf is also full of self pride, which is his fatal flaw. All of these factors combined make it impossible for Beowulf to be anything less than a show off. He has much to gain: respect, his rightful place as King, and another notch in his belt of pride.
Oh, absolutely! What could cement his reputation more than fighting the beast bare handed? All the greatest Danish warriors had become cheese for Grendel's Triscuits, even with weapons.
Furthermore, one of the criteria for proof that one is indeed an epic hero (according to Joseph Campbell's classic study, The Hero With a Thousand Faces) is that:
Ultimately the hero or heroine must rely on his or her own strength, wits and resources to emerge victorious.