Calvinists love to act like they have a monopoly on Monergism. If you are not a Calvinist, according to Calvinists, then you must logically hold to a Synergistic view of salvation. This actually has implications for the Gospel itself, and so Calvinists really need to consider the implications of this false claim. In effect, what they are saying is you are either a Calvinist, or a Gospel denier.
Of course, to many people, the terms "Monergism" and "Synergism" are meaningless. They're two of those fancy theological terms that don't actually come from the Bible, nor are they typically spoken about in your average church. So, what are they?
Monergism literally means "to work alone", and as that implies, it means that God is the sole worker in our salvation. This, of course, describes Calvinism quite nicely, as it effectively teaches that mankind are just puppets in salvation. Not that any Calvinist will ever admit that, and indeed they tend to hate the analogy, but given that in their worldview, God makes us unable to obey Him, then if He happens to want us saved, He makes us unable to resist Him, how else can it be described?
Monergism is a Biblical certainty. Biblically speaking, we are saved by grace, through faith, as the gift of God, not of works, lest any man should boast (Ephesians 2:8-10). When we get to Heaven, none of us will be able to claim "I helped God get me here". What we earned for ourselves is eternal wrath. God is responsible for 100% of our salvation.
Synergism, by contrast, means "to work together". It is the belief that some contribution, however small, must be made in order to bring about our salvation. Synergism is Biblically unsustainable.
The debate on whether one can be a non-Calvinist and remain a Monergist, ultimately, hinges on whether or not faith is a work. If faith is a work, then by believing in the name of the Son of God unto salvation, you have somehow contributed to your own salvation. If, however, faith is not a work, then one can believe salvation is dependent upon your faith (as Scripture repeatedly affirms), and not be a Synergist.
To illustrate this even before opening the Bible, I like to point out that God has done a great many more works than just salvation. As a Christian, I believe all truth proceeds from God. All reality, aside from God Himself (for He is necessary), is either a passive or active result of God's will. I do not believe God causes sin, because James 1:13-14 tells me that He simply cannot do so. Furthermore, God explicitly tells us that some things that did happen were not in His will (e.g. Jeremiah 7:31), and that we can resist His will (e.g. Luke 7:30).
Thus, we must conclude that some things, God merely permits, all in pursuit of a higher purpose. This is why even in Romans 9:22, God is said to endure with much longsuffering the vessels of wrath, not that He intentionally made them children of wrath.
But on the other hand, everything good, everything pure, everything that is even potentially corrupted by the stain of sin, exists because of His will. This includes the very existence of those who bear His image. Adam, when he was created, was "very good", as was his wife, Eve. These were made by the will of God. Similarly, though we are sinners, each of us have, from the beginning, been creations of God.
But not everyone is wise enough to believe that. Some would rather trust in asinine fairy tales, such as Evolution. Some are wise enough to reject that, but still refuse to confess and glorify God as their Creator. But I am not among them. I wholeheartedly confess that every breath I take comes from the Lord. Now, tell me, how much credit do I get for my life if I only believe God has given it to me? The only sane answer is none. I did not give myself life, I cannot prolong my own life, I cannot even give it up until the Lord Himself tells me "no more". Now, if my faith in the Author of my life gets me no credit for this life, which is destined to end, how can it possible get me credit for the next life, which does not? It's Monergism! God does it all!
But you don't even need to think of it from a purely logical point of view, because Scripture itself distinguishes between faith and works. Of course, there is the obvious example of Ephesians 2:8-10, which not only tells us we are saved by grace, through faith, not of works, and even goes as far as to tell us a few details about how God prepared good works for us to walk in beforehand.
But there's also Romans 4:5, which tells us that he who does not work, but believes on Him who justifies the unGodly, his faith is credited to him for righteousness. So, is faith a work? No. If it was, Scriptures like these would effectively be saying "we are not saved by works, but by works", or "to him who does not work, but works", which are obviously illogical statements.
If faith is not a work, then having faith does not automatically make salvation Synergistic. It is ridiculous to believe otherwise. In fact, I invite Calvinists to test this themselves. Next time you receive a gift, attempt to claim credit for it. I promise you, if you still have that friend afterwards, you will not be receiving as many gifts from them.
All of the work for salvation was performed before most of the recipients were even born. By whom? By the Lord. It was He who created the heavens and the Earth. It was He who set Adam and Eve in the Garden of Eden. It was He who commanded them not to eat, yet it was they who ate. It was He who promised them a Savior, who chose patriarchs and raised up kings in His line, who chose the blessed virgin to bring Him forth. It was He who walked the Earth, preaching "repent, for the Kingdom of Heaven is at hand". It was He who chose the Apostles, one of them a devil who would betray Him. It was He who lay down His life on the cross, and it was He who took it right back up again. It was He who preserved the record of all of this in documents He inspired. It was He who ensured these documents would endure, and would spread. It was He who ensured you would be born in exactly the right place and time, your very biology being in His hand. It was He who ensured the message would reach your ears, and that every necessary circumstance for you to believe was actualised.
"But you did believe, so you contributed". What? The irony is, unless you deny that faith is the requirement for salvation, at which point you deny the Gospel itself, you must believe that you also believe, and that is why you are saved. It wouldn't be much better to say this than to say God empowers you to work for your salvation.
Ultimately, the Gospel is an entirely Monergistic phenomenon. God does all the work, and gets all the credit. You do not have to be a Calvinist to be a Monergist. You simply have to believe Scripture.