For many, there are specific texts in scripture that are gateway drugs to specific doctrines. For me, Romans 16 and Judges 4 were both a gateway to adopting an egalitarian reading of scripture. For others who are interested in the Reformed/Calvinist and Arminian/Wesleyan debate (a debate between brothers and sisters of good will), John 6, Hebrews 6, 2 Peter 3, and Romans 9 are often considered the central prooftexts in this debate, although there are many other considerations. For some of my Reformed brothers and sisters, however, John 6:44 is considered the mainstay text. Jonathan Dorst at The Chorus in the Chaos blog on Patheos writes
As I began to study Calvinism, this was the thread that wove throughout: that salvation is a work of God from first to last. I saw that, though we are responsible for our actions and sin, and though the outward call is universal (“whoever comes to me I will never cast out”), God is the prime mover in saving His people. God is not up in heaven wringing His hands over who will choose Him, and He is not casting a vote that gets equal influence with the devil’s vote hoping to win our patronage- God is actively drawing people to Himself. Here in John 6:44 was a hint of the effect of total depravity, the implication of unconditional election, the inescapable conclusion of limited atonement, the stark reality of irresistible grace, and a building block for the doctrine of the perseverance of the saints (which, of course, is about God persevering to perfect His people and to raise them up on the last day, not us working to stay in God’s good graces)…. We all believe the Bible, but we interpret it differently, and we need the help of godly men and women who have gone before us to understand the Bible. And while John Calvin and his disciples were gifted, but flawed, theologians, and Calvinism is just a tradition and is not perfect, it is the tradition that I believe represents Scripture most accurately. And John 6:44 was my gateway drug to becoming a Calvinist.
Personally, I find this sort of theological method and journey fascinating. In many respects, when a person reads a particular text or book is almost more important then what they read. This is not to make light of Dorst's comments or look down upon people who have aligned themselves to a specific theological group with a clear conscience. Rather, the time of much of our reading and research is almost as important as what we are researching. Just a thought on that. I also cite Dorst not to refute him (although I do not think John 6:44 is helpful to Calvinism in context at all), but to simply illustrate the interpretive power at work for many people within a specific Christian tradition: who we read—whether Calvin or Wesley or Beza or Spurgeon or Arminius or Oden—often determines which specific texts gain our hermeneutical imagination. The seeds of a specific worldview are often planted before we ever turn to Holy Scripture.
Audience in John 6
John uses two specific terms for the audience surrounding Jesus. He first uses the phrase "a great crowd" (πολὺς ὄχλος) throughout the beginning pericopes. In John 6:2 and 6:5, throughout the feeding of the five thousand, the "great crowd" does not leave Jesus but "follows him" (ἠκολούθει: imperfect verb). The pericopes in John 6 may be divided into the "feeding of the give thousand" discourse (vv.1-15), the "walking on water" discourse (vv.16-25) and the "bread of life" discourse that occurs for the rest of the chapter (vv.26-71). John seems to single out what could be called Jewish opposition (οἱ Ἰουδαῖοι) in 6:4 and 6:41 as well, which suggests that Jesus' comments in the Gospel of John are concerned with his Jewish interlocutors.
The Heavenly Son, Grumbling, and John 6:41-43
The imperfect verb ἐγόγγυζον ("grumbling": v.41, 61; see also ἐμάχοντο in v.52) is our first indication of the mood of Jesus' interlocutors (οἱ Ἰουδαῖοι). These Jewish leaders, hardly representative of all Jewish people in the crowds, are concerned with Jesus' claims to "descend from heaven" (ὁ καταβὰς ἐκ τοῦ οὐρανοῦ), implying preexistence in some form. The context, therefore, does not appear to be about Calvinism or Arminianism, but on the origin of the Son (c.f. 1:1-18). A heavenly figure descending from heaven is not a foreign concept necessarily in Judaism (Dan 4:13, 23 LXX), but given the apocalyptic imagery of Daniel, one cannot necessarily fault the Jewish leaders for not seeing the obvious.
God's Sovereignty and Human Faith in John 6:44-51
How a non-Calvinist will understand vv.44-51 (and vv.61-66 by implication) can be largely reduced to how we exegete certain words. Those words include the negated participle δύναται ("able"), the verb πιστεύω ("to believe, have faith") throughout John 6, and the aorist ἑλκύσῃ ("draw"). I will address these in order.
On the first verbal phrase οὐδεὶς δύναται, we must be clear about what the phrase does not say. The phrase does not specify exactly for what reason one is "not able" to come to the Son. Dorst (and many of my Reformed brothers and sisters) have to supply a reason for this inability (i.e. total depravity, which I affirm but do not see as the reason) but the text itself seems to provide a specific reason. Specifically, v.45 uses the adjectival phrase διδακτοὶ θεοῦ ("learners of God," or "God's learners" depending on how one interprets the genitive θεοῦ) to speak about those who "hear/ understand" (ἀκούσας), which is based upon the knowledge received from God. As a consequence of this learning and understanding, a person can then come to the Son. But, as with much of the New Testament and Jewish thought, the concept of learning requires participation in what one has learned. I have written on this elsewhere. That is, having learned and understood, one is then required to "come" (ἔρχεται: middle voice, suggesting personal agency) to the Son as a consequence of adopting and participating. One is unable to come to God without learning about what God requires. The universal witness of God is for all people (πάντες) and is predicated upon active participation in God's call. The cognitive element of this learning and understanding cannot be stressed enough.
Thus, a person's inability to come to God may be conditioned on total depravity, but God's universal prevenient grace draws us to him regardless.
This flows nicely into John's use of the verb πιστεύω (6:29, 30, 35, 36, 40, 47, 64 [2x], 69). The verb refers to "believe something to be true and, hence, worthy of being trusted - 'to believe, to think to be true, to regard as trustworthy (Louw-Nida). Faithfulness is a precondition that demands a person's awareness of the Son and the Father, and an active sense of participation in the mission of the Spirit. As Jesus says in 6:47, "Amen, amen, I say to you, the one who believes (i.e. exercises faithfulness) has eternal life" (ἀμὴν ἀμὴν λέγω ὑμῖν, ὁ πιστεύων ἔχει ζωὴν αἰώνιον). John Wesley in his Explanatory Notes on the New Testament, says this concerning John 6:44:
No [person] can believe in Christ, unless God gives him power; he draws us first, by good desires. Not by compulsion, not by laying the will under any necessity; but by the strong and sweet, yet still resistible, motions of his heavenly grace.
The final consideration concerns the verb ἑλκύω. In the New Testament, this verb does not appear to be used primarily in a soteriological or eschatological context (John 18:10; 21:6, 11; Acts 16:19; 21:30; James 2:6), save for John 12:32. If John 12:32 is interpreted in the way of "to drag" or "compel," then one ends up with universalism in some form. While some may insist on distinctions in how they understand "all" in that verse, I do not find such arguments compelling—but that is another debate for another time. Suffice to say, the power of the Son to resurrection (ἀναστήσω: 6:44b) is predicated upon the exercise of human faithfulness: resurrection to eternal life (as opposed to destruction per 3:16) is conditioned on human participation in the life of the Spirit. Marianne Meye Thompson argues that the verb ἑλκύω most likely means, "to attract." She writes
In John the emphasis on God's love for the world argues strongly for [the aforementioned meaning of "to attract"]. According to Jer 31:3 (38:3 LXX), because God loves Israel with an eternal love, God has drawn them…with compassion; later Jeremiah promises that God himself will write the law on the hearts of his people so that they no longer need teachers…that prophetic vision comes to fruition in God's drawing people to Jesus.
In summation, I respect the different positions many take in interpreting John 6:44. Personally, I believe the reasons I have provided above offer non-Calvinists a more consistent way to understand this wonderful text:
God's glory is manifested in the eternal Son, and all are called to learn and faithfully participate in the mission of God for the reconciliation of the world.
To God be the glory.
 Jonathan Dorst, "John 6:44—The Verse that Made me a Calvinist," http://www.patheos.com/blogs/chorusinthechaos/john-644-the-verse-that-made-me-a-calvinist/.
 See here: http://www.splitframeofreference.com/blog-1/2017/12/15/learning-in-the-pastoral-epistles-deception-verbs-and-wives-in-1-timothy-2
 Marianne Meye Thompson, John: A Commentary (New Testament Library. Louisville: Westminster John Known, 2015), 152-153.