Because that is how crucifixion killed.
That's one of many possible ways to die by crucifixion. I don't believe the texts support that he was suffering from asphyxiation. Here's what some scholars say on the matter. I know of none that say the sour wine made it harder for him to breath making death by asphyxiation happen faster.
Gill:
John 19:29
Now there was set a vessel full of vinegar,.... In a place near at hand, as Nonnus observes; not on purpose, for the sake of them that were crucified, either to refresh their spirits, or stop a too great effusion of blood, that they might continue the longer in their misery; but for the use of the soldiers who crucified Christ, vinegar being part of the allowance of Roman soldiers (m), and what they used to drink: sometimes it was mixed with water; which mixed liquor they called "Posca" (n), and was what even their generals sometimes used; as Scipio, Metellus, Trajan, Adrian, and others: vinegar was also used by the Jews for drink, as appears from Rth_2:14 and "dip thy morsel in the vinegar", which Boaz's reapers had with them in the field; "because of heat", as the commentators say (o); that being good to cool, and to extinguish thirst; for which reason the soldiers here offer it to Christ; though the Chaldee paraphrase of the above place makes it to be a kind of sauce or pap boiled in vinegar; and such an "Embamma" made of vinegar the Romans had, in which they dipped their food (p); but this here seems to be pure vinegar, and to be different from that which the other evangelists speak of, which was mingled with gall, or was sour wine with myrrh, Mat_27:34. Vinegar indeed is good to revive the spirits, and hyssop, which is after mentioned, is an herb of a sweet smell; and if the reed, which the other evangelists make mention of, was the sweet calamus, as some have thought, they were all of them things of a refreshing nature: vinegar was also used for stopping blood (q), when it flowed from wounds in a large quantity; and of the same use were sponges; hence Tertullian (r) mentions "spongias retiariorum", the sponges of the fencers, which they had with them to stop any effusion of blood that should be made in their exercises; but then it can hardly be thought that these things should be in common prepared at crucifixions for such ends, on purpose to linger out a miserable life a little longer, which would be shocking barbarity; and especially such a provision would never be, made at this time, on such an account, since the Jews sabbath drew nigh, and they were in haste to have the executions over before that came on, that the bodies might not remain on the cross on that day; for which reason they would do nothing, at this time, however, to prolong the lives of the malefactors; wherefore it is most reasonable, that this vessel of vinegar was not set for any such purpose, but was for the use of the soldiers; and therefore this being at hand when Christ signified his thirst
Barnes:
That the scripture might be fulfilled, saith, I thirst - See Psa_69:21. Thirst was one of the most distressing circumstances attending the crucifixion. The wounds were highly inflamed, and a raging fever was caused, usually, by the sufferings on the cross, and this was accompanied by insupportable thirst. See the notes at Mat_27:35. A Mameluke, or Turkish officer, was crucified, it is said in an Arabic manuscript recently translated, on the banks of the Barada River, under the castle of Damascus. He was nailed to the cross on Friday, and remained until Sunday noon, when he died. After giving an account of the crucifixion, the narrator proceeds: “I have heard this from one who witnessed it; and he thus remained until he died, patient and silent, without wailing, but looking around him to the right and the left, upon the people. But he begged for water, and none was given him; and the hearts of the people were melted with compassion for him, and with pity on one of God’s creatures, who, yet a boy, was suffering under so grievous a trial. In the meantime the water was flowing around him, and he gazed upon it, and longed for one drop of it; and he complained of thirst all the first day, after which he was silent, for God gave him strength” - Wiseman’s Lectures, pp. 164, 165, ed.
John 19:29
A vessel full of vinegar - This was probably that tart small wine which we are assured was the common drink of the Roman soldiers. Our word vinegar comes from the French vin aigre, sour or tart wine; and, although it is probable that it was brought at this time for the use of the four Roman soldiers who were employed in the crucifixion of our Lord, yet it is as probable that it might have been furnished for the use of the persons crucified; who, in that lingering kind of death, must necessarily be grievously tormented with thirst. This vinegar must not be confounded with the vinegar and gall mentioned Mat_27:34, and Mar_15:23. That, being a stupifying potion, intended to alleviate his pain, he refused to drink; but of this he took a little, and then expired, Joh_19:30.
