I think the argument from Scripture is much stronger than our cited theologian gives, but this summarizes my efforts with 𝑴𝒆𝒍𝒄𝒉𝒊𝒛𝒆𝒅𝒆𝒌 𝒂𝒏𝒅 𝒕𝒉𝒆 𝑳𝒂𝒔𝒕 𝑺𝒖𝒑𝒑𝒆𝒓. I think it should be of considerable importance for Protestant inquirers (especially those who respect tradition), not only because of the Scriptural evidence (which is sufficient), but also the unanimity of the Church Fathers.
“Finally, something must be said of the argument to be drawn from Christ’s priesthood ‘according to the order of Melchisedech.’ The argument, as repeated in dozens of theological text books, many be thus briefly set down. Priesthood and sacrifice are correlative; priests of the same order must offer sacrifice according to the same rite. Melchisedech offered sacrifice in bread and wine, therefore so did Christ. But the only time he can possibly be said to have done this was at the Last Supper, and therefore the Eucharist is a sacrifice.
Intrinsically and as a purely scriptural argument this 𝗺𝗮𝘆 𝘀𝗲𝗲𝗺 to be defective. The Greek word translated ‘order’ refers rather to rank, quality, manner, than to the sacrificial rite. To this no reference seems to be made either in the Psalm or in the Epistle; in the latter the writer is wholly occupied with the eternity and superiority of Christ’s priesthood as compared with that of Aaron. This he illustrates and explains by saying that Christ is ‘a priest according to the order of Melchisedech.’ The King of Salem is shown to be Abraham’s superior by receiving from him the tribute of tenths; he is the type of the eternity of Christ’s priesthood by his manner of appearing in the pages of Scripture, ‘without father, without mother, without geneaology, having neither beginning of days nor end of life,’ and therefore he is ‘likened unto the Son of God (and) continueth a priest for ever.’ Hence those who are content with the purely objective and apparently obvious interpretation of Scripture may reject the argument.
𝗕𝘂𝘁 𝘁𝗵𝗲 𝗖𝗮𝘁𝗵𝗼𝗹𝗶𝗰 𝗵𝗮𝘀 𝗮𝗻𝗼𝘁𝗵𝗲𝗿 𝗰𝗿𝗶𝘁𝗲𝗿𝗶𝗼𝗻; 𝗳𝗼𝗿 𝗵𝗶𝗺 𝘁𝗵𝗲 𝗖𝗵𝘂𝗿𝗰𝗵 𝗶𝘀 𝘁𝗵𝗲 𝗼𝗻𝗹𝘆 𝗮𝘂𝘁𝗵𝗼𝗿𝗶𝘁𝗮𝘁𝗶𝘃𝗲 𝗶𝗻𝘁𝗲𝗿𝗽𝗿𝗲𝘁𝗲𝗿 𝗼𝗳 𝗛𝗼𝗹𝘆 𝗪𝗿𝗶𝘁, 𝗮𝗻𝗱 𝗵𝗲𝗿 𝘃𝗼𝗶𝗰𝗲 𝘀𝗽𝗲𝗮𝗸𝘀 𝗶𝗻 𝘁𝗵𝗲 𝗰𝗼𝗻𝘀𝘁𝗮𝗻𝘁 𝘁𝗿𝗮𝗱𝗶𝘁𝗶𝗼𝗻 𝗼𝗳 𝗵𝗲𝗿 𝗙𝗮𝘁𝗵𝗲𝗿𝘀 𝗮𝗻𝗱 𝗗𝗼𝗰𝘁𝗼𝗿𝘀. 𝗟𝗼𝗼𝗸𝗲𝗱 𝗮𝘁 𝗶𝗻 𝘁𝗵𝗶𝘀 𝗹𝗶𝗴𝗵𝘁 𝘁𝗵𝗲 𝘄𝗼𝗿𝗱𝘀 𝘂𝗻𝗱𝗲𝗿 𝗿𝗲𝘃𝗶𝗲𝘄 𝗮𝗽𝗽𝗲𝗮𝗿 𝗮𝘀 𝗮 𝗰𝗼𝗻𝘃𝗶𝗻𝗰𝗶𝗻𝗴 𝗽𝗿𝗼𝗼𝗳 𝗼𝗳 𝘁𝗵𝗲 𝘀𝗮𝗰𝗿𝗶𝗳𝗶𝗰𝗶𝗮𝗹 𝗰𝗵𝗮𝗿𝗮𝗰𝘁𝗲𝗿 𝗼𝗳 𝘁𝗵𝗲 𝗟𝗮𝘀𝘁 𝗦𝘂𝗽𝗽𝗲𝗿, 𝗳𝗼𝗿, 𝗳𝗿𝗼𝗺 𝘁𝗵𝗲 𝗯𝗲𝗴𝗶𝗻𝗻𝗶𝗻𝗴 𝗼𝗳 𝘁𝗵𝗲 𝘀𝗲𝗰𝗼𝗻𝗱 𝗰𝗲𝗻𝘁𝘂𝗿𝘆 𝗼𝗻𝘄𝗮𝗿𝗱𝘀, 𝗵𝗮𝗿𝗱𝗹𝘆 𝗮 𝗖𝗵𝗿𝗶𝘀𝘁𝗶𝗮𝗻 𝘄𝗿𝗶𝘁𝗲𝗿 𝗾𝘂𝗼𝘁𝗲𝘀 𝘁𝗵𝗲𝗺 𝘄𝗶𝘁𝗵𝗼𝘂𝘁 𝘀𝗲𝗲𝗶𝗻𝗴 𝗶𝗻 𝘁𝗵𝗲𝗺 𝗮 𝗿𝗲𝗳𝗲𝗿𝗲𝗻𝗰𝗲 𝘁𝗼 𝗖𝗵𝗿𝗶𝘀𝘁’𝘀 𝗶𝗻𝘀𝘁𝗶𝘁𝘂𝘁𝗶𝗼𝗻 𝗼𝗳 𝘁𝗵𝗲 𝗘𝘂𝗰𝗵𝗮𝗿𝗶𝘀𝘁 𝗮𝗻𝗱 𝗮 𝗱𝗲𝗺𝗼𝗻𝘀𝘁𝗿𝗮𝘁𝗶𝗼𝗻 𝗼𝗳 𝘁𝗵𝗲 𝘀𝗮𝗰𝗿𝗶𝗳𝗶𝗰𝗶𝗮𝗹 𝗰𝗵𝗮𝗿𝗮𝗰𝘁𝗲𝗿 𝗼𝗳 𝘁𝗵𝗲 𝗠𝗮𝘀𝘀. 𝗔𝘀 𝗣𝗲𝘁𝗮𝘃𝗶𝘂𝘀 𝗽𝘂𝘁𝘀 𝗶𝘁:: ‘𝑶𝒏 𝒕𝒉𝒊𝒔 𝒑𝒐𝒊𝒏𝒕 𝒕𝒉𝒆 𝒂𝒏𝒄𝒊𝒆𝒏𝒕 𝒘𝒓𝒊𝒕𝒆𝒓𝒔 𝒂𝒈𝒓𝒆𝒆 𝒕𝒐 𝒔𝒖𝒄𝒉 𝒂𝒏 𝒊𝒏𝒄𝒓𝒆𝒅𝒊𝒃𝒍𝒆 𝒆𝒙𝒕𝒆𝒏𝒕, 𝒕𝒉𝒂𝒕 𝒕𝒉𝒆𝒓𝒆 𝒄𝒂𝒏 𝒃𝒆 𝒏𝒐 𝒓𝒐𝒐𝒎 𝒇𝒐𝒓 𝒍𝒆𝒈𝒊𝒕𝒊𝒎𝒂𝒕𝒆 𝒅𝒐𝒖𝒃𝒕 𝒊𝒏 𝒕𝒉𝒆 𝒎𝒊𝒏𝒅 𝒐𝒇 𝒂𝒏𝒚 𝑪𝒉𝒓𝒊𝒔𝒕𝒊𝒂𝒏.'”
“The Eucharistic Sacrifice,” in Rev. B. V. Miller, D.D., Ph.D., in The Teaching of the Catholic Church: A Summary of Catholic Doctrine, ed. Canon George D. Smith, D.D., Pd.D., (New York: The Macmillan Company, 1955), 885-886.