Adamo Tadolini's statue of St. Paul stands in front of St. Peter's Basilica at the Vatican. (Photo credit: ‘AngMoKio’, CC BY-SA 2.5, via Wikimedia Commons)
“I have written to you very boldly by way of reminder, because of the grace given me by God to be a minister of Christ Jesus to the Gentiles in the priestly service of the gospel of God...” (Romans 15:15-17)
The future universality of the priesthood is clearly indicated in the Old Testament:
Malachi 1:11 (RSV) For from the rising of the sun to its setting my name is great among the nations, and in every place incense is offered to my name, and a pure offering; for my name is great among the nations, says the LORD of hosts. (cf. Is 66:18, 21; Jer 33:14-22; Zeph 3:9-10)
St. Paul casually assumes that priests are still operative under the new Christian covenant, by referring to the table of the Lord (or altar) and contrasting it with the table of demons, in a eucharistic context:
1 Corinthians 10:14-21 Therefore, my beloved, shun the worship of idols.  I speak as to sensible men; judge for yourselves what I say.  The cup of blessing which we bless, is it not a participation in the blood of Christ? The bread which we break, is it not a participation in the body of Christ?  Because there is one bread, we who are many are one body, for we all partake of the one bread.  Consider the people of Israel; are not those who eat the sacrifices partners in the altar?  What do I imply then? That food offered to idols is anything, or that an idol is anything?  No, I imply that what pagans sacrifice they offer to demons and not to God. I do not want you to be partners with demons.  You cannot drink the cup of the Lord and the cup of demons. You cannot partake of the table of the Lord and the table of demons. (cf. 9:13)
The apostle Paul is in this same priestly thought-world in another of his utterances:
Romans 15:15-17 But on some points I have written to you very boldly by way of reminder, because of the grace given me by God  to be a minister of Christ Jesus to the Gentiles in the priestly service of the gospel of God, so that the offering of the Gentiles may be acceptable, sanctified by the Holy Spirit.  In Christ Jesus, then, I have reason to be proud of my work for God.
He’s “offering” a “priestly service” to the Gentiles. The Greek word is hierourgeo: Strong’s Concordance defines it as “to be a temple-worker, i.e., officiate as a priest (fig.): -- minister.” This classic (non-Catholic) reference work states: “to minister in the manner of a priest, minister in priestly service.” It also notes (from Joseph Thayer’s Greek-English Lexicon) historical etymological definitions of “to be busied with sacred things; to be perform sacred rites” (from Philo), and “used esp. of persons sacrificing” (from Josephus).
Baptist Greek scholar A. T. Robertson, in his famous work, Word Pictures of the New Testament (Nashville, Tennessee: Broadman Press, 1930; six volumes; under Romans 15:16; vol. 4, 520), provides the basic definition: “to work in sacred things, to minister as a priest.” Likewise, Marvin Vincent's Word Studies in the New Testament (four volumes; New York: Charles Scribner’s Sons, 1887; reprinted: Grand Rapids, Michigan: Wm. B. Eerdmans Publishing Co., 1946; vol. III, 174) states, for the same passage:
Ministering (ierourgounta). Only here in the New Testament. Lit., ministering as a priest.
Offering up (prosfora). Lit., the bringing to, i.e., to the altar. Compare doeth service, John xvi. 2.
Kittel’s Theological Dictionary of the New Testament defines it as:
‘to perform sacred or sacrificial ministry.’ In Josephus and Philo it always means “to offer sacrifice” and often has no object. (hierourgia means “sacrifice” and hierourgema the “act of sacrifice.”)
None of these reference works are Catholic; thus, no charge of bias based on Catholic affiliation can be made against them. The bottom line is that Paul has called himself a priest – using two different terms.
We get the word liturgy from litourgos (Strong’s word #3011; cf. #3008, 3009, and 3010). Strong’s (word #3008: litourgeo) applies it to, among other things, “priests and Levites who were busied with the sacred rites in the tabernacle or the temple.”
The author of Hebrews applies one of these terms to priests in the old covenant sense in the following passages:
Hebrews 9:21 And in the same way he sprinkled with the blood both the tent and all the vessels used in worship. (cf. 10:11)
Likewise, Jesus is a “high priest”:
Hebrews 8:1-2 Now the point in what we are saying is this: we have such a high priest, one who is seated at the right hand of the throne of the Majesty in heaven,  a minister in the sanctuary and the true tent which is set up not by man but by the Lord.
Given the central motif in the New Testament of Jesus as the sacrificial lamb, it stands to reason that the Sacrifice of the Mass would be associated with the Eucharist, as the central rite of Christian worship.
St. Paul also casually assumes the continued existence of altars among Christians (1 Cor 10:14-21), and altars are mentioned in the New Testament in other places (apart from the many mentions of altars in heaven), as well:
Hebrews 13:9-12 Do not be led away by diverse and strange teachings; for it is well that the heart be strengthened by grace, not by foods, which have not benefited their adherents.  We have an altar from which those who serve the tent have no right to eat.  For the bodies of those animals whose blood is brought into the sanctuary by the high priest as a sacrifice for sin are burned outside the camp.  So Jesus also suffered outside the gate in order to sanctify the people through his own blood.
Therefore, if it is true that – as John Calvin argues in his Institutes: IV, 18:3 –: “the cross of Christ is overthrown the moment an altar is erected”, then the New Testament is against the cross. It’s much more likely that Calvin has misunderstood the passages we have been analyzing.