What is a very simple way to determine whether the Apostle Paul was Calvinistic or Arminian in his belief?
I would contend that Paul was neither, given that neither theologian was born at the time of Paul's writings, and that Joseph Arminius was only a fraction removed from supporting Calvin's premises. In fact, the only place they disagreed was Calvin's belief that predestination was based on God's choosing some for salvation and not others, whereas Arminius believed it was based on God's foreknowledge of who would believe. I wonder if you realize that Arminius did
not believe one could lose his/her salvation? Yet that is an attribute of the so-called "Arminians" of today. It is wrong to collectively deposit anyone who does not accept the the Calvinist view of predestination in the Arminian camp, because the vast majority who reject that view actually embrace eternal security.
Now, that said, your view of Romans 9-11 ignores a very key factor in the exegesis and analysis of the three chapters. To grasp what these often misinterpreted chapters are about, recall the major theme of Romans: God's unending faithfulness to his promises. God is faithful to us in spite of our unfaithfulness to him. While we still were sinners, Christ died for us. While we still were God's enemies, we were reconciled to God through the death of his Son (5:8-10). But, Paul now asks, what about unfaithful Israel?
Does Israel's unfaithfulness mean that Israel has dropped out of God's plan? Does it mean that God has rejected Israel as the people through whom he will bless the world? Has the word of God failed? Else how can it be that the very people whose entire history had been looking forward to the Messiah's coming have now rejected him? Paul's answer is this: Not so. God's promises to Israel once made are not retracted under the impact of Israel's rejection of the Messiah. God has not rejected Israel, nor will God ever do so. The Jews continue to be God's chosen people and to play a role in his plan to save the world. Here is where I part company quite emphatically with covenant theology, and contend that Calvinists who adopt covenant belief have abandoned Calvin.
Paul has no explanation for the hardening that has come over Israel. He doesn't understand why it came about. Israel's rejection of Jesus is a mystery. Israel's rejection is not the same as the rejection by unbelievers, which is to be found everywhere. It is a special rejection, and must be respected as such. In his Israel chapters, Romans 9 to 11, Paul saw God's will in Israel's rejection. It is not because Israel says "No" to Christ that Israel's heart has been hardened. It is because God hardened its heart that it cannot do anything but say "No." Hardness of heart is not the same thing as rejection. It is an act for a particular purpose.
Israel's rejection of the gospel is actually God's own doing. It serves to spread of the gospel among the Gentile nations. Despite turning its back on their own Messiah, Israel remains God's beloved on account of the promises God made to the patriarchs in Romans 11.
Romans 11, NASB
1 I say then, God has not rejected His people, has He? May it never be! For I too am an Israelite, a descendant of Abraham, of the tribe of Benjamin.
2 God has not rejected His people whom He foreknew. Or do you not know what the Scripture says in the passage about Elijah, how he pleads with God against Israel?
3 "Lord, THEY HAVE KILLED YOUR PROPHETS, THEY HAVE TORN DOWN YOUR ALTARS, AND I ALONE AM LEFT, AND THEY ARE SEEKING MY LIFE."
4 But what is the divine response to him? "I HAVE KEPT for Myself SEVEN THOUSAND MEN WHO HAVE NOT BOWED THE KNEE TO BAAL."
5 In the same way then, there has also come to be at the present time a remnant according to God's gracious choice
The heart of Romans 9-11 is Paul's metaphor of the olive tree.
Romans 11
16 If the first piece of dough is holy, the lump is also; and if the root is holy, the branches are too.
17 But if some of the branches were broken off, and you, being a wild olive, were grafted in among them and became partaker with them of the rich root of the olive tree,
18 do not be arrogant toward the branches; but if you are arrogant, remember that it is not you who supports the root, but the root supports you.
19 You will say then, "Branches were broken off so that I might be grafted in."
20 Quite right, they were broken off for their unbelief, but you stand by your faith. Do not be conceited, but fear;
21 for if God did not spare the natural branches, He will not spare you, either.
22 Behold then the kindness and severity of God; to those who fell, severity, but to you, God's kindness, if you continue in His kindness; otherwise you also will be cut off.
23 And they also, if they do not continue in their unbelief, will be grafted in, for God is able to graft them in again.
24 For if you were cut off from what is by nature a wild olive tree, and were grafted contrary to nature into a cultivated olive tree, how much more will these who are the natural branches be grafted into their own olive tree?
The tree, both roots and branches, is Israel. The broken-off branches are the Jews who have rejected the gospel of Jesus Christ. Their rejection is not permanent but temporary. God in his good time will graft the natural, broken-off branches back into "their own olive tree" (v. 24).
The grafted branches are the gentiles who have accepted the gospel. Gentiles who believe in Jesus do not become part of Israel, just as a twig from a wild olive tree grafted into a cultivated olive tree. It receives the but rather are fed by the root, sustained by the "food and water" of the root system. That root system is not Israel, but Christ, who would not be known apart from Israel. In case you were unaware, a wild olive branch grafted into a cultivated tree
will never produce an olive, but instead, only serve to sustain the tree through its leaves, collecting sunlight and breathing in carbon dioxide to fuel photosynthesis. Did Paul know that at the time he wrote? I couldn't answer. But the reality of this grafting is unmistakable.
Therefore, we, as the grafted in branches, keep the gospel alive, keep it growing and sustaining the other branches that may be grafted in, including Jews who will eventually come to know Christ, thus being grafted back into their own tree.
For many centuries the Church has been contemptuous of the Jews, claiming that God rejected the Jews and replaced them with the Church. This contempt derailed the Church's thinking about God. Its thinking was this: because Israel rejected Jesus the Messiah, God rejected Israel. But that line of thinking leads to a blind alley. Unless God proves faithful to disobedient Israel, what reason is there to believe that God will be faithful to the disobedient Gentiles? The God who rejects the disobedient people he once chose and then instead chooses another people is not the kind of God who will save a disobedient world.