They claim the government has admitted personal information about families cannot be shared under the scheme in any way which would break the law.
Simon Calvert, a Christian and spokesperson from the No To Named Person group, said: "However they try to spin it, this is a major climbdown by the Scottish Government."
Under the proposals every person under the age of 18 in Scotland would have a designated adult - usually a health or teaching professional - who would be a point of contact for anyone who had concerns for a child's welfare.
A named person would be responsible for helping support and advise families in the event that concerns regarding wellbeing were identified.
The Scottish Education Secretary John Swinney announced new legislation was being brought forward which would require a professional acting as a named person to consider whether sharing details about a child or family would be illegal.
He also said they should "consider whether sharing information would promote, support or safeguard the wellbeing of the child or young person''.
The Supreme Court ruled last summer that the data-sharing elements of the scheme were "incompatible" with the privacy and family life rights set out in the European Convention on Human Rights (ECHR).
Dr Gordon Macdonald from the Christian charity CARE in Scotland said: "The named-person scheme has been strongly criticised by both parents and leading professionals across all spheres of life - education, health and law enforcement.
"The Scottish Government must listen to the majority and not shoehorn in a proposal that is deeply unpopular and serves to undermine the role of parents in family life."