Michael Gove has said that as prime minister he would make it clear that primary schools teaching children about LGBT relationships is a government requirement.
The former education secretary said doing this would send a clear message that any protests should be aimed at the government and not headteachers on the frontline.
However, he said parents can and should have the right to withdraw pupils from sex education.
LGBT lessons in primary school have sparked months of protests in Birmingham, with one school – Anderton Park Primary – now at the heart of a legal battle over whether demonstrations can continue outside its gates.
Schools facing LGBT protests
Some headteachers and the NAHT headteachers' union have called on ministers to do more to support schools facing these protests.
Mr Gove said: “The intimidating protests outside schools in Birmingham are unacceptable and must stop. I support the High Court injunction to stop them.
“While parents can and should have a right to withdraw their children from sex education, that does not apply to the wider curriculum.
“As education secretary, I introduced the mandatory teaching of British values, to make sure young people were taught about the importance of tolerance and respect for people who are different to them.
“I believe that teaching children that some people have two mums and two dads is an important part of preparing them for life in modern Britain. Indeed, many children in primary schools will have classmates with same-sex parents.
"While it should be for primary schools to decide how to introduce this teaching as part of the curriculum, I will be clear that teaching the topic is a government expectation. In doing so, we will send a clear message that any protests or disagreement should be directed at the government, not individual headteachers on the frontline.”
His comments have been published on the LGBT+ Conservatives Group website.
Conservative leadership candidates have set out their views, policies and plans for LGBT rights.
The government is introducing compulsory relationship education for primary schools and relationship and sex education for secondary schools from 2020.
The NAHT said the guidance for primary schools, published by the Department for Education, says relationship education will include LGBT content, but it wants a stronger message of support to back this up.
Earlier this week, the NAHT’s general secretary, Paul Whiteman, raised concern about education secretary Damian Hinds telling heads that primary schools “are enabled and encouraged to cover LGBT content if they consider it age-appropriate to do so”.
Mr Whiteman said: “We’d like the ‘if’ changed to ‘when’ " to make it clear that primary schools are right to teach children about LGBT people."