Does John 6:37 confirm 'irresistible grace'?
Short answer: No.
Basic Assessment of Syntax / Semantics
Πᾶν ὃ δίδωσίν μοι ὁ Πατὴρ πρὸς ἐμὲ ἥξει καὶ τὸν ἐρχόμενον πρός με οὐ μὴ ἐκβάλω ἔξω ὅτι καταβέβηκα ἀπὸ τοῦ οὐρανοῦ οὐχ ἵνα ποιῶ τὸ θέλημα τὸ ἐμὸν ἀλλὰ τὸ θέλημα τοῦ πέμψαντός με Τοῦτο δέ ἐστιν τὸ θέλημα τοῦ πέμψαντός με ἵνα πᾶν ὃ δέδωκέν μοι μὴ ἀπολέσω ἐξ αὐτοῦ ἀλλὰ ἀναστήσω αὐτὸ ἐν τῇ ἐσχάτῃ ἡμέρᾳ τοῦτο γάρ ἐστιν τὸ θέλημα τοῦ Πατρός μου ἵνα πᾶς ὁ θεωρῶν τὸν Υἱὸν καὶ πιστεύων εἰς αὐτὸν ἔχῃ ζωὴν αἰώνιον καὶ ἀναστήσω αὐτὸν ἐγὼ ἐν τῇ ἐσχάτῃ ἡμέρᾳ
All that the Father giveth to me shall come to me; and him that cometh to me, I will not cast out. Because I came down from heaven, not to do my own will, but the will of him that sent me. Now this is the will of the Father who sent me: that of all that he hath given me, I should lose nothing; but should raise it up again in the last day. And this is the will of my Father that sent me: that every one who seeth the Son, and believeth in him, may have life everlasting, and I will raise him up in the last day.
It is very clearly talking about an objective group: those who meet criteria x will be y—those who shall have ended up being such as I have described (x) will have y happen to them.
In the context of the Bible—in the context of John—this is those predestinate in the sense that God doesn't 'already' know who will ultimately be saved, He simply knows, tenseless, all things, in an eternal present.
We know it's an objective group (and thus the 'predestined' in the sense that God foreknew these saved individuals would be saved) because "All that the Father giveth to me", "he that" etc. are not specific and have no particular referent; they are a type of people, i.e. a certain group: the elect.
Jesus' words could be paraphrased as "I know that those who are saved were drawn by the Father and that I did not lose any of them (these are definitionally not those who I did not save); these ones I saved." If He were speaking in the past tense.
Notice He says, "And this is the will of my Father that sent me: that every one who seeth the Son, and believeth in him, may have eternal life". Thus, the believer plays an active role, an excercise of will. But clearly from the Gospels and the New Testament in general, it is God who is the author of salvation, not man, who merely agrees its a good deal and co-operates to recieve the benefits of New Testament salvation.
No man can come to me, except the Father, who hath sent me, draw him; and I will raise him up in the last day.
Notice this very important verse (v. 44). It identifies the Father as He who draws the believer: it does not indicate that all He draws will come to Him. Notice how "that of all that he hath given me, I should lose nothing" does not say He drew only them. The rest of the New Testament is clear "[God Our Saviour] wishes for all men to be saved and to come to a knowldege of the truth." (1 Timothy 2:4; cf. Ezekiel 33:11; Matthew 23:37 etc).