State-funded Holyport College near Maidenhead, opened to pupils for the first time in September.
The Queen and the Duke of Edinburgh were given a tour of the school before unveiling a plaque watched by its 122 pupils, staff and parents.
Holyport College, sponsored by Eton College, accepts boarders and gives its pupils access to some of Eton's facilities.
During the visit, the royal couple met Sir Nicholas Winton, who lives near the school which has named its reception building after him.
Sir Nicholas, dubbed the "British Schindler", helped to save hundreds of children, mainly from Jewish families, from the Nazis by transporting them by train from Prague to the UK in 1939.
The wheelchair-bound 105-year-old said: "I think it's a wonderful honour to have the building named after me.
"I hope the school will make contact with a school in the Czech Republic that is also named after me."
Other guests included Home Secretary Theresa May, MP for Maidenhead.
The school was proposed in February 2012 and then approved by the Department for Education through the free schools programme in June 2012.
Once it reaches capacity, the school will have 500 pupils, including 225 boarders aged 11-19 years.
Parents pay £3,850 a term for boarders but daytime education is free.
Free schools, which anyone can apply to set up, are funded directly by central government and are free from local council control.
A free school in Berkshire has been officially opened by the Queen.