"Irresistible grace" is a notion that seems to have originated with Augustine of Hippo (354-430), who took a rather extreme view against Pelagius (c 360-418), a Briton who emphasized the superiority of free will over grace. Augustine went too far in the other direction. In his treatise, "On Rebuke and Grace", for example, he writes:
Will you dare say that even when Christ prayed that Peter's faith might not fail, it would still have failed if Peter had willed it to fail? As if Peter could in any measure will otherwise than Christ had wished for him that he might will."1
Other Church Fathers of the time criticized this view (while perhaps not identifying Augustine by name) and even Augustine himself later recanted some of his more extreme views. John Cassian (365-435), for example, wrote:
AND so these are somehow mixed up and indiscriminately confused, so that among many persons, which depends on the other is involved in great questionings, i.e., does God have compassion upon us because we have shown the beginning of a good will, or does the beginning of a good will follow because God has had compassion upon us? For many believing each of these and asserting them more widely than is right are entangled in all kinds of opposite errors. For if we say that the beginning of free will is in our own power, what about Paul the persecutor, what about Matthew the publican, of whom the one was drawn to salvation while eager for bloodshed and the punishment of the innocent, the other for violence and rapine? But if we say that the beginning of our free will is always due to the inspiration of the grace of God, what about the faith of Zaccheus, or what are we to say of the goodness of the thief on the cross, who by their own desires brought violence to bear on the kingdom of heaven and so prevented the special leadings of their vocation?
These two then; viz., the grace of God and free will seem opposed to each other, but really are in harmony, and we gather from the system of goodness that we ought to have both alike, lest if we withdraw one of them from man, we may seem to have broken the rule of the Church's faith: for when God sees us inclined to will what is good, He meets, guides, and strengthens us: for "At the voice of thy cry, as soon as He shall hear, He will answer thee;" and: "Call upon Me," He says, "in the day of tribulation and I will deliver thee, and thou shalt glorify Me."2 And again, if He finds that we are unwilling or have grown cold, He stirs our hearts with salutary exhortations, by which a good will is either renewed or formed in us.3
I think this background is relevant because it is hard to imagine that the Church Fathers, who opposed any doctrine of irresistible grace, would have opted to include the Gospel of John in the canonical New Testament Scriptures as they did at the local council of Carthage in 397 and the later 7th Ecumenical Council in 787, if said Gospel would have been understood to contain teachings contrary to their doctrine.
Notwithstanding, neither is their support for such an interpretation in the Gospel text itself. You quote only v.37, but the full context includes v.35-36:
Jesus said to them, “I am the bread of life; whoever comes to me shall not hunger, and whoever believes in me shall never thirst. But I said to you that you have seen me and yet do not believe. All that the Father gives me will come to me, and whoever comes to me I will never cast out.
Jesus does not say, "whoever the Father gives to me shall never thirst", but rather, "whoever believes in me shall never thirst." The conclusion to be drawn here is that those whom the Father gives are whoever believes. One Greek commentary on the passage explains:
All that the Father giveth Me shall come to Me
This means, "The Father gives Me those who believe in Me. You Jews, being unworthy, are not given to Me by the Father. This is why you do not come to Me: you are not upright of heart. If you were, God the Father in His love for you would lead you to faith in Me.4
An earlier Greek commentary by John Chrysostom (c 349-407) reads:
And in this place, by the “which the Father giveth Me,” He declareth nothing else than that “the believing on Me is no ordinary thing, nor one that cometh of human reasonings, but needeth a revelation from above, and a well-ordered soul to receive that revelation.”
But perhaps some one will say ... if none can come unto Thee except it be given him from above, then those to whom the Father giveth not are free from any blame or charges.” These are mere words and pretenses. For we require our own deliberate choice also, because whether we will be taught is a matter of choice, and also whether we will believe.5
1. Seraphim Rose, The Role of Blessed Augustine in the Orthodox Church (St. Herman's Press, 2007), p.37
2. Isaiah 30:19; Psalm 49:15 LXX
3. Conferences XIII.11
4. Theophylact, Explanation of the Holy Gospel According to St. John (tr. from the Greek; Chrysostom Press, 2007), p.106
5. Homily XLV on John