No, not all Christians recognize Peter as the first pope or head of the Church. This is a point of significant difference between various Christian denominations.
* Catholic Church: The Catholic Church believes that Jesus appointed Peter as the first pope and head of the Church. They base this belief on passages like Matthew 16:18-19, where Jesus says, "And I tell you, you are Peter, and on this rock I will build my church, and the gates of hell shall not prevail against it. I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven." Catholics interpret this as Jesus giving Peter a unique authority that was then passed down through the line of popes.
* Eastern Orthodox Church: The Eastern Orthodox Church recognizes Peter as an important apostle and holds him in high esteem. However, they do not believe that Peter held a position of supreme authority over the other apostles or that his authority was passed down to a single successor (the pope). They believe in the concept of "primacy of honor" for the Bishop of Rome, but not "primacy of jurisdiction" or infallibility.
* Protestant Churches: Most Protestant denominations do not recognize Peter as the first pope. They interpret the passages in Matthew differently, often arguing that the "rock" Jesus referred to was Peter's confession of faith in Jesus as the Christ, rather than Peter himself. They also believe that all believers have equal access to God and that there is no single earthly head of the Church. Authority is typically seen as residing in Scripture, and in some denominations, in the collective leadership of pastors or elders.
In summary, the recognition of Peter as the first pope is a distinctly Catholic belief, with varying views held in Eastern Orthodox and Protestant traditions.