Under the law, every child in Scotland will be assigned a state-employed named person to monitor their welfare.
Critics argue that assigning a single point of contact, such as a teacher or health visitor, to look out for the welfare of children aged under 18 breaches data protection laws and the human rights of parents.
It is due to be introduced in August but Christian charity CARE wants it scrapped or at least redrafted.
CARE for Scotland Parliamentary Officer Dr Gordon Macdonald said: "The Scottish Government's named person scheme is very dangerous and will undermine parents and their role as the best guardians of their children.
"Simple logic dictates if you spread resources too thin, which this scheme will inevitably do, vulnerable children who are most in need of help may well be overlooked and put at risk.
"We have brought this case to the UK Supreme Court because it has the power to overrule the Scottish Government and also because we remain utterly convinced this scheme breaches international human rights laws and therefore needs to be scrapped or redrafted."