When we consider what a Byronic hero is, we need to recall how Lord Byron, a man who was described as "mad, bad and dangerous to know," exemplifies this figure. A Byronic hero then is a flawed, dark and jaded character, weary of the world and something of an outsider. However, crucial to a Byronic hero is his ability to attract and draw in women who are irresistibly impelled towards them because of their badness.
When we consider the character of Gabriel Conroy, then, it is clear that he is anything but a Byronic hero. He is desperate to fit in and he does not know how to handle women or his audience. He is obsessed with his level of education and is concerned, that, for example, if he quotes Browning in his speech, his audience will not understand the reference. In a sense, he is a very passive character, especially compared to his foil in the story, Michael Furey, who is an active, strident figure, his name suggests, compared to the complacence and lack of self-confidence exhibited by Gabriel. As the end of the story makes clear, the figure of Michael Furey and the love that he had for his wife stands as a sharp rebuke for Gabriel's lack of depth of love:
He thought of how she who lay beside him had locked in her heart for so many years that image of her lover's eyes when he had told her that he did not wish to live.
Thus Gabriel Conroy can definitely not be described as a Byronic hero. In fact, he is depicted as almost the opposite, but a figure nonetheless who experiences an epiphany about himself and the human condition at the end of the story.