I think that one of the most Romantic characteristics of Martin Fierro is that he remains distinct from the conformist society, at large. Prior to his disposition in the army, he exists on the Pampas, singing his song and loving his family. In this, one sees his Romantic tendencies in that he is a part of the land, willing to live a life that works for what he values as he does not seek to become something he is not. The torment of the Romantic hero, almost Byronic in scope, is evident when Martin Fierro returns to a home destroyed and wife and children gone. The fact that he sets out on a quest of revenge against those who did him wrong is Romantic because he is able define his own sense of being, without negotiation from a social setting or willingness from other external forces to define him. Martin's crusading ways for justice is another example of the Romantic hero because he is relentless in his mission. There is something Romantic about Martin roaming the countryside, on the hunt for his family, and being almost "beyond" the reach of anyone or anything else. The fact that Martin's narrative is one that is sung is another example of how the Romantic hero is personified. He sings his narrative. Although it is filled with suffering and pain, it is a song to be sung. This accentuates the aesthetic in human experience, a very Romantic quality.
It's important to note that Latin American Romanticism has very different characteristics from European Romanticism. The themes linked to Latin American Romanticism in El gaucho Martin Fierro are:
Sentimental love, history, Native peoples, Black people, political freedom, American nature, nostalgia for a homeland
The reproduction of peasant speech, Americanisms, regionalisms, indigenous and African voices