Beneath his bushy eyebrows, two small piercing eyes darted about the room as he spoke, giving him the look of a scavenging buzzard.
Word's grammar checker is dysfunctional; turn it off or ignore it. Grammatically, that's a perfectly good sentence, not a fragment. It does read quite awkwardly, though. I think you're trying to cram too much into one sentence. I can't find any order for it that flows smoothly. Every variation I've tried has something in an awkward place.
I'm afraid, however, that my issues with this sentence go far beyond grammar.
"Eyes darted about the room" makes me visualize a pair of eyeballs jumping out of his head and bouncing around the room. Have you considered "eyes glanced", "eyes peered", or "gaze darted"?
"Two small piercing eyes" is quite a pile-up of adjectives. Most people have two eyes; why is it important to tell us this guy does, too? Even so, "two" is the least noticeable. (ETA: I just reread the thread. You've already changed "two" to "his". Doesn't help much, though.)
The "small piercing" combo is what really gets me. Plus, there are two more adjectives elsewhere in the sentence. The overall effect is cloying.
And maybe it's just me, but "bushy eyebrows" aren't very suggestive of a "scavenging buzzard".