The shadow health secretary said schools that use religion as an excuse to avoid teaching about gay and lesbian relationships were "straightforwardly wrong".
"We refute Andy Burnham's claims that Catholic schools do not teach about same-sex marriage. Catholic schools teach factually about all aspects of the law and emphasise the importance of strong and positive relationships in both primary schools and secondary schools," said a CES statement to Premier.
"Both the Education Select Committee's Report 'Life lessons: PSHE and SRE in schools' and OFSTED's examples of good practice use Catholic schools as an example of outstanding SRE. This is a testimony to the hardworking teachers across the country who are at the forefront of passing on our Catholic and British values.
Mr Burnham also said blamed the Government for allowing some sex education to become watered down through free schools, which he suggested had allowed homophobic bullying to develop.
"That has to be tackled right there, and not only should SRE be absolutely compulsory, but there must be absolute equality in terms of all relationships within Sex and Relationships Education in terms of how it is taught," Andy Burnham told Pink News.
"I worry about an education policy that is making the academy or the free school judge and jury, letting it basically do what it wants," added Burnham.
This was also refuted by the body representing 2,300 Catholic schools, colleges and universities: "Catholic schools, like all schools, should have a zero tolerance to homophobic bullying, as with all other types of bullying. We expect our Catholic schools to be places of safety and learning for all pupils. Every pupil in our schools ought to feel valued as a unique part of God's creation, made in the image and likeness of God," read the Catholic Education Service statement.