John Calvin's commentary on this passage affirms that "water" means the Spirit:
Though the name Water is borrowed from the present occurrence, and applied to the Spirit, yet this metaphor is very frequent in Scripture, and rests on the best grounds. For we are like a dry and barren soil; there is no sap and no rigour in us, until the Lord water us by his Spirit. In another passage, the Spirit is likewise called clean water, (Heb 10:22,) but in a different sense; namely, because he washes and cleanses us from the pollutions with which we are entirely covered. But in this and similar passages, the subject treated of is the secret energy by which he restores life in us, and maintains and brings it to perfection.
There are some who explain this as referring to the doctrine of the Gospel, to which I own that this appellation is fully applicable; but I think that Christ includes here the whole grace of our renewal; for we know that he was sent for the purpose of bringing to us a new life. In my opinion, therefore, he intended to contrast water with that destitution of all blessings under which mankind groan and labor. Again, living water is not so called from its effect, as life-giving, but the allusion is to different kinds of waters. It is called living, because it flows from a living fountain.
He finds that the reference to the spring or fountain has a particular significance:
For he does not say that, from the very first day, we drink so as to be fully satisfied, but only means that the Holy Spirit is a continually flowing fountain; and that, therefore, there is no danger that they who have been renewed by spiritual grace shall be dried up. And, therefore, although we thirst throughout our whole life, yet it is certain that we have not received the Holy Spirit for a single day, or for any short period, but as a perennial fountain, which will never fail us. Thus believers thirst, and keenly thirst, throughout their whole life; and yet they have abundance of quickening moisture; for however small may have been the measure of grace which they have received, it gives them perpetual vigor, so that they are never entirely dry. When, therefore, he says that they shall be satisfied, he contrasts not with Desire but only with Drought.
Shall be a fountain of water springing up into eternal life. These words express still more clearly the preceding statement; for they denote a continual watering, which maintains in them a heavenly eternity during this mortal and perishing life. The grace of Christ, therefore, does not flow to us for a short time, but overflows into a blessed immortality; for it does not cease to flow until the incorruptible life which it commences be brought to perfection.
The final paragraph shows the relevance of this passage to Calvinist ideas of irresistible grace and perseverance of the saints.
To me, this text brings to mind Matthew 20:20-23 and Mark 10:35-40, where Jesus asks his disciples "Can you drink the cup that I am to drink?". This question has been variously interpreted as referring to:
- General troubles and pains of life, not necessarily martyrdom
- The Eucharist
- Receiving the Holy Spirit
- Living a Christ-like life
It may not be possible or desirable to separate these meanings. Calvin's commentary is happy to assign multiple interpretations to the cup, since they are all bound up in the same idea.