In a new analysis, First Focus said the shift would "ration" healthcare coverage for Medicaid's 30 million children.
"Per capita caps do nothing to improve access to or the quality of health care for our nation’s most vulnerable citizens," First Focus President Bruce Lesley said. "Instead, they will only serve to shift the federal share of costs to states, low-income senior citizens, people with disabilities, and children."
The group's analysis said the policy would also create a complicated bureaucracy, and pose a special threat for children with severe illnesses.
Hatch, in January, touted the idea for its history of traction in both political parties. "Bipartisan per-capita cap reforms would implement desperately needed fiscal discipline in Medicaid while preserving access to care for beneficiaries," Hatch's office wrote.
First Focus received attention in 2012 when it came out against Rep. Paul Ryan's (R-Wis.) House budget. That plan would block-grant Medicaid, another popular idea within the GOP.
Children's advocates said the policy would lead to "rationing and outright denials for kids" on Medicaid.