Dusten Brown had earlier signed a legal document agreeing to put the girl up for adoption, but his attorneys say the father did not understand the extent of the waiver, and that the birth mother misrepresented the child's American Indian heritage to social service workers when the adoption was finalized.
At issue is whether Brown, as the onetime non-custodial father, can gain parental custody, after the non-Indian mother initiated an adoption outside the tribe.
A special congressional law governs such interstate adoptions, since the current 556 federally recognized tribes all fall under Interior Department oversight, giving those tribes certain unique benefits and rights.
Lawyers for the Capobiancos say federal law does not define an unwed biological father as a "parent."
The adoptive couple was excited that the high court will hear their case.
"We weren't sure what to expect," Melanie Capobianco told CNN's Randi Kaye. "It was a low chance and we just feel really extremely happy that they decided to hear it."
Her husband, Matt, added, "It restored some hope and a little faith in the judicial system."
The federal law in question is the Indian Child Welfare Act (ICWA) of 1978, a response to decades of often abusive social service practices that resulted in the separation of large numbers of native youngsters from their families, in many cases to non-Indian homes.
The legislation was designed to "promote the stability and security of Indian tribes and Indian families by the establishment of minimum federal standards to prevent the arbitrary removal of Indian children from their families and tribes and to ensure that measures which prevent the breakup of Indian families are followed in child custody proceedings."
Brown's relationship within the "federally recognized government" of the Cherokee Nation means Veronica -- named in court papers as "Baby Girl" -- is a member of the tribe and subject to their jurisdiction.
"It's not anyone's intent ever to rip a child away from a loving home," said Todd Hembree, the Tahlequah, Oklahoma-based tribe's attorney general. "But we want to make sure those loving homes have the opportunity to be Indian homes first."
Still, the Capobiancos argue that the little girl's real home is with them.
Image: The U.S. Supreme Court, via Shutterstock