The latest Ofsted report for the school rated it outstanding in all areas
A Fertile Heart is being taught at St Mary's Roman Catholic High School, in Herefordshire, through The Archdiocese of Cardiff.
The county council said it was "at odds" with the role of encouraging children to respect differences.
The archdiocese has been contacted by the BBC for a response.
The programme also says men were "created to initiate sexual relationships" and women are to be "receiver-responders".
Felicity Norman, Herefordshire Council's children and families cabinet member, said she had serious concerns about it, although the local authority had "no power" over what the archdiocese chose to teach within county schools.
"It seems to be at odds with the essential role of a school to foster caring and co-operative relations between all children and staff, to respect differences and to support and encourage children as they negotiate the difficulties of adolescence," she said.
"We are disturbed at the failure of the Archdiocese of Cardiff, responsible for directing its schools as to what it teaches, to respond to us over this or other matters concerning the safeguarding of children, in spite of attempts on our part to engage with them."
* The programme came about after the announcement of the new Relationships, Sex and Health Education (RSHE) curriculum being made mandatory in UK schools in 2017
* It provides faith-based RSHE education to Catholic schools and overseas, "to ensure the quality of content and that the delivery is in line with Catholic teachings"
* It is already run in all 56 Catholic schools in Cardiff
* The strategy is supported by the Archbishop of Cardiff and the Bishop of Shrewsbury, and the archbishop has recommended the resource to the Congregation for Catholic Education at the Vatican
Stuart Wetson, head teacher of the school in Lugwardine, told the Local Democracy Reporting Service St Mary's was a Catholic school and its relationships and sex education policy was inspired by religious teachings of the Roman Catholic faith.
It was acting in accordance with government guidance on relationships and sex education (RSHE) and was "committed to providing our pupils with opportunities for broad and balanced debate on a range of issues whilst teaching distinctive faith perspectives on relationships", he said.
"We continue to review and evaluate our practice, sharing feedback from teachers, pupils and parents with the Archdiocese of Cardiff and the publishers of key resources."
The latest Ofsted report for the school in 2019 rated it outstanding in all areas and praised pupil development and its "extremely effective" safeguarding.
The Equality Act 2010 means schools cannot unlawfully discriminate against pupils because of their sex, race, disability, religion or belief or sexual orientation.
Lucy Whitehouse, from online sex education platform, Fumble, said it had "never been more important" for schools to teach pupils about sex and relationships properly.
"The government is giving schools room to be creative with it which is important, particularly for faith schools... but there needs to be clear lines about ideas of gender and LGBTQ," she added.
"To suggest contraception is wrong and gay and lesbian people cannot marry and must abstain from sex, everything there I would be quite happy to affirm is wrong.
"It is a backward step.. and LGBTQ pupils will be listening to that in school and it's harmful and leads to lasting damage."
In November, the education secretary was asked by Labour MP Stella Creasy whether material published on the website A Fertile Heart was permitted to be used in schools according to government guidance.
Schools minister Nick Gibb said it was for schools to decide which resources they chose to support the teaching of RSHE.
He said there was clear advice for schools on choosing resources within the Department for Education's "Plan your relationships, sex and health curriculum", as well as the statutory RSHE.
"The department has stated that schools should not work with agencies that take extreme positions, and this should also be reflected in the school's choice of resources," he said.
In a statement, a spokesperson for the programme said those behind A Fertile Heart were "sad" to hear councillors' concerns.
"The programme A Fertile Heart contains 115 lessons that have been written in line with Catholic values and everything that Pope Francis has taught us.
"These lessons prioritise love, tolerance, human dignity and respect," it said.
If the grass is greener on the other side, you can bet the water bill is higher.