How is Beowulf in the epic tale Beowulf not chivalrous?
There are very few things about Beowulf's main character that are not chivalrous.
[C]hivalry is essentially a martial ideal, a code of values that glorified military prowess as the supreme achievement of the virtuous knight.
The characteristics of the virtuous knight, elements of behavior and character that offset a knight's brutal nature in battle make up a long list.
The qualities of the Chivalric knight were summarized in Raymond Lull’s Book on The Order of Chivalry (ca. 1270). Some of those qualities are to be:
...able-bodied, of good lineage, wise, generous, loyal, courageous, and honorable.
Ethically, the chivalrous knight must be willing to:
...defend the Christian faith, defend his lord, to protect the weak, exercise constantly by hunting and jousting in tournaments, pursue robbers and evil-doers.
And such a knight must avoid:
...pride, lechery, false oaths, and treachery.
Beowulf appears to meet all of these requirements. He is able-bodied, as seen when he fights Grendel:
The foul bandit took a mortal wound, and a fatal tear appeared on his shoulder. His sinews ripped apart and his bone-frame broke. Victory was now given to Beowulf, and Grendel, sick unto his death, went hence and sought his den in the dark moors...
Beowulf is wise, loyal, courageous, and honorable as seen in the way he prepares his men to battle the monster and does not fear losing his life. He is loyal to his clan's leader, Hygelac, and honorable in his dealings with Hrothgar—coming to the Geatish king's aid. He is also polite in asking Hrothgar for his permission that only Beowulf and his men face Grendel:
And so I seek from you, sovereign of the glorious Danes, bulwark of the Scyldings, a boon—and, friend of the peoples, shield of the warriors, do not refuse it now that I've come from afar—that I alone, with my liegemen here, this stalwart band, may purge Heorot!
When Unferth questions his bravery in Beowulf's battle with Breca, the Geatish hero states the facts of their contest, but he is not prideful or boastful. Beowulf pursues evildoers, such as Grendel and Grendel's dam (mother).
We might infer that Beowulf defends the Christian faith as he credits God with power to destroy all evil:
But God is able to halt the deeds of this deadly fiend!
The only way that I find Beowulf not to be chivalrous by definition of a medieval description of knights is that he does not...
...exercise constantly by hunting and jousting in tournaments.
He goes to battle for the good of his clan and others (e.g., the Geats). However, he does not seem to do anything in the way of hunting or jousting. While we often associate these things with entertainment, nothing about Beowulf speaks to participation in anything entertaining. He is a sober and serious warrior for the Danes.
Rather than not being a chivalric warrior, Beowulf would appear to be a fine example of a man who is chivalrous beyond question.