The main battlefield in the Calvinism debate is Romans 9. Because of the content of this particular chapter, it's easy to see why one might be persuaded by Calvinism, and it is because of this that I simply cannot consider Calvinism a heresy. I don't believe it's true, but I also don't believe it's so false that it can be dismissed without due care.
Verses 19-21 are particularly popular among Calvinists: "You will say to me then, “Why does He still find fault? For who has resisted His will?” But indeed, O man, who are you to reply against God? Will the thing formed say to him who formed it, “Why have you made me like this?” Does not the potter have power over the clay, from the same lump to make one vessel for honor and another for dishonor?"
Taking Calvinism to its logical conclusion, "Why does He still find fault? For who has resisted His will?" is a valid objection. If it is truly God's will that we sin, and thus we sin because He compels us to do so, then it is, in fact, God who is at fault. "Who are you to reply against God?" Well, apparently, someone whom God designed to reply against God.
So we see that actually, the Calvinist interpretation of Romans 9 renders the verse illogical. And of course, it is contrary to other Scriptures. 1 Timothy 2:1-4 clearly tells us that God desires all men to be saved and come to the knowledge of the truth, and this is why we are to pray for them. Ezekiel 18:32 and 33:11 both clearly tell us God takes no pleasure in the death of the wicked. James 1:13-14 tells us God cannot tempt anyone, but we are tempted when we are drawn away by our own desire. The same author, in the same book, tells us that God has shown unrighteous men, who suppress the truth in unrighteousness, what may be known of Him, so they are without excuse (Romans 1:18-22).
When all Scriptures are taken into account, the Calvinist interpretation of Romans 9 cannot be sustained, because it gives unbelievers the very excuse the Bible says they cannot have. But why, they may ask, does it seem to so heavily suggest that it is God's will to destroy the wicked, even forming them for this very purpose?
There is a relatively simple solution. I say relatively because this is a highly complex topic; nothing about it is simple. However, there are other beliefs that would raise the objection "why does He still find fault? Who has resisted His will?" and would still merit the same reply: "Who are you to reply against God?"
A key element in this discussion that Calvinists never seem to acknowledge is God's foreknowledge. God's foreknowledge, according to Romans 8:29, plays a direct role in how He predestines the elect. "For whom He foreknew, He also predestined to be conformed to the image of His Son, that He might be the firstborn among many brethren." (Emphasis mine).
As much as Calvinists seem to hate the idea that God knew every possible future before He began to act, atheists, ironically, seem to have a reasonable grasp on it. For this reason, they will often argue that God is responsible for every person who ever goes to Hell. Why? Well, because He knew who would go to Hell, yet He created them anyway.
Thus, with little to no knowledge of Calvinism, atheists make the very objection we see in Romans 9: "Why does He still find fault? Who has resisted His will?" The answer remains the same: "Who are you to reply to God?" But unlike the Calvinist interpretation, an atheist has no grounds to reply. They cannot continue to say "well, apparently, someone whom God made to reply to Him". No, they are now faced with the fact that they have a command to repent (Acts 17:30), which God absolutely desires them to obey (1 Timothy 2:4), and is patiently giving them time to obey because He isn't willing that they should perish (1 Peter 3:9), and yet, maybe they will, maybe they won't.
See, God wins either way. If a sinner repents, God is glorified by His mercy. If a sinner becomes hardened in his ways, God is glorified by executing justice. Justice, because the sinner has no excuse for either the sins he freely committed, nor for the rejection of grace.
But the Calvinist might answer "but it says God has mercy on whom He wills, and whom He wills, He hardens". To that, I say let's go back to Pharaoh. Notice, in this account, both God and Pharaoh receive credit for hardening Pharaoh's heart. In Exodus 8:15, 32, and 9:34 Pharaoh, rather than God, is said to have hardened his own heart. God tells Moses, even before this happened, that He knew Pharaoh would not let the Jews go unless forced by a heavy hand (Exodus 3:19).
And so the "simple" answer is that there is no way for God to lose. Man cannot foil God, as if our acceptance of the Gospel He chose to deliver to us is somehow overwhelming Him. As if our condemnation is depriving Him, who can make children for Abraham from the very stones (Matthew 3:9; Luke 3:8), of anything. We see, then, that while Romans 9, studied in isolation, can seem to be a powerful case for Calvinism, but when studied in light of other Scriptures, it soon becomes clear that it does not say what Calvinists so often insist it does. It is not the claim that the God who cannot deny Himself (2 Timothy 2:13) denies Himself by tempting people to sin without tempting them to sin (James 1:13-14). It is not the claim that the God who wants all men to be saved (1 Timothy 2:4) and no one to perish (2 Peter 3:9) actually made some people specifically for the purpose of perishing rather than being saved. It is not the claim that the God in whom there is no darkness (1 John 1:5) is so full of darkness that He wills for people to resist His will (Luke 7:30). No, Romans 9 is a description of God's complete and utter control over world history, even being able to use the sins we commit of our own free will to accomplish His perfect plan, even if that means He knew, even before He chose which actions to take, what effect those actions would have.
In the end, God finds fault in the vessels of wrath because there is fault in the vessels of wrath. They are clay which, rather than conforming to His original design, becomes marred in His hands. They do things which never entered His heart, and when He calls them to repent, they reject His will (Luke 7:30). To their peril, not to His. Therefore, Calvinism can be safely rejected, simply because it is the only view that gives unbelievers the excuse the Bible says they don't have.