Beowulf's only weaknesses are his ego and pride, which lead him to recklessly take on challenges alone. This sets him apart from Greek and Roman heroes, who always have a tragic flaw that leads to their destruction.
After a night of boasting, Beowulf takes on the dreaded monster Grendel and then dives deep into a pool after Grendel's deadly mother. Beowulf does these things without fear, sometimes even without weapons, with the confidence that he will succeed.
Even when he is older and weaker, he tackles a dragon by himself. When he is mortally wounded in defeating the dragon, he accepts the fate that decreed he would die at that point. This separates Beowulf from the Greek classical heroes, whose fates were caused by their own bad choices at critical moments. Beowulf must fight the dragon or accept the death of many. In fighting it, he demonstrated Western fatalism, rather than Greek hamartia.