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Protestants would say that Papal Supremacy is not anywhere outlined in Scripture. As a rebuttal to the Catholic interpretation of Matt 16:18, you will sometimes see Protestants make the Petros/Petra distinction.
Greek Orthodox, on the other hand, acknowledge that Peter was the first Pope, that he was appointed Pope by Christ, and they even acknowledge that the Pope is first among the brother bishops. However, they object to the doctrine of papal supremacy on the grounds that they do not believe that the tradition of the Church permits the pope to make unilateral declarations on matters of doctrine outside of what the Catholic Church would call the "ordinary" magisterium. In their conception of papal primacy, the Pope cannot have any power to exercise over other bishops. From a Greek article on the subject
If primacy is defined as a form of power, then we encounter the question of whether in the Orthodox church there is a power superior to that of a bishop, i.e., a power over the bishop, and hence the church of which he is head. Theologically and ecclesiologically the answer must be an unconditional no: there is no power over the bishop and his church. In the canonical and historical life of the Church, however, such supreme power not only exists but is conceived as the foundation of the Church; it is the basis of its canonical system. According to Father Schmemann, this reflects the alienation of canonical tradition from ecclesiology and its reduction to canon law in the context of which the life of the Church came to be expressed in juridical terms.