The Church's ruling General Synod endorsed a report apologising for the abuse of several children by two priests in the Chichester diocese in the 1970s and 80s.
The cases of Roy Cotton and Colin Pritchard prompted an inquiry by the Archbishop of Canterbury's office into safeguarding procedures in the diocese.
Members also unanimously backed an earlier apology issued by the Archbishops of Canterbury and York.
The Bishop of Southwell and Nottingham, the Right Reverend Paul Butler, said the church's failures were “sin, just as much as the perpetrators sinned.”
The bishop said the Church of England "failed big time" over child protection.
Rt Rev Butler said for "far too long" the Church of England, notably those in senior positions, had either disbelieved the stories of victims, believed them but tried to hide the truth away or hoped that by removing an offender the problem would go away.
"We failed to listen properly, we did not acknowledge the wrong done, and we protected the institution at the expense of the person abused."
His remarks followed a 30-second silence observed by the General Synod after the bishop read a statement from the Stop Church Child Abuse Group (SCCA), a coalition of survivor support groups, to the General Synod.
In the statement the SCCA called for an independent public inquiry into the extent of child abuse within the Church of England.
The Rt Rev Butler said the Chichester report exposed "serious failures" in the diocese but also "much wider institutional failings" which affect every diocese in the Church of England.
In a strongly worded statement calling for the public inquiry, the SCCA said: "If the Church through this Synod is willing to walk with us on this costly journey then there may be a purpose and there may be hope for survivors and for the Church in this resolution and this apology.”
The General Synod also gave its backing for a programme of changes designed to tighten up child protection procedures in the Church of England and prevent further scandals.
The proposals include removing the 12-month limit for bringing complaints under the Clergy Discipline Measure for complaints alleging sexual abuse.
Clergy who have been defrocked or suspended, or who have no licence or permission to officiate, would also be prevented from robing or wearing clerical vestments in church under the proposals.