Independent Fundamental Churches deny the Roman Catholic interpretation of Matthew 16:18. In their reading, the rock upon which the church is built is either Christ himself or the confession of Christ's deity—not Peter's person or office. The passage does not establish papal supremacy or apostolic succession, which fundamentalists view as unscriptural innovations.

This interpretation reflects the fundamentalist separation from the institutional Catholic Church and the emphasis on local church autonomy and congregational authority. Each local church is accountable directly to Christ.