As a Christian, can I just let go of my abusive family?
No. I will presume here that you are not speaking of a spouse or children but of your family of origin. For better or for worse, your family will always be your family, and you should always seek to remain in some kind of relationship with them.
This doesn't mean that you have to be in direct contact with them, especially if abuse is ongoing or if it is presenting a danger to immediate family members, such as children, for whom you are responsible. But neither should you cut yourself off from your family. If nothing else, they deserve your prayers for their conversion. If it becomes necessary, they also should be able to depend on you for assistance—for instance, should they become seriously ill.
Whatever else your family of origin has done, your parents gave you life, and it is a basic Christian obligation to honor that gift by honoring our parents and extended family. Jesus reiterated this injunction when he told the Pharisees, "For God commanded, 'Honor your father and your mother,' and, 'He who speaks evil of father or mother, let him surely die'" (Matt. 15:4). The Catechism of the Catholic Church states:
The fourth commandment is addressed expressly to children, in their relationship to their father and mother, because this relationship is the most universal. It likewise concerns the ties of kinship between members of the extended family. It requires honor, affection, and gratitude toward elders and ancestors. Finally, it extends to the duties of pupils to teachers, employees to employers, subordinates to leaders, citizens to their country, and to those who administer or govern it (2199).