Here John Calvin’s discussion of the phrase “the washing of regeneration” in Titus 3:5 from his commentary. I will post some paragraphs from his sermon on this passage later.

I have no doubt that he alludes, at least, to baptism, and even I will not object to have this passage expounded as relates to baptism; not that salvation is contained in the outward symbol of water, but because baptism seals to us the salvation obtained by Christ. Paul treats the exhibition of the grace of God, which, we have said, has been made by faith. Since therefore a part of revelation consists in baptism, that is, so far as it is intended to confirm our faith, he properly makes mention of it. Besides baptism-being the entrance into the Church, and the symbol of our ingrafting into Christ-is here appropriately introduced by Paul, when he intends to shew in what manner the grace of God appeared to us; so that the strain of the passage runs thus:-‘God hath saved us by his mercy, the symbol and pledge of which he gave in baptism, by admitting us into his Church, and ingrafting us into the body of his Son.

Now the Apostles are wont to draw an argument from the Sacraments, to prove that which in there exhibited under a figure, because it ought to be held by believers as a settled principle, that God does not sport with us by unmeaning figures, but inwardly accomplishes by his power what he exhibits by the outward sign; and therefore baptism is fitly and truly said to be ‘the washing of regeneration.’ The efficacy and use of the sacraments will be properly understood by him who shall connect the sign and the thing signified, in such a manner as not to make the sign unmeaning and inefficacious, and who nevertheless shall not, for the sake of adorning the sign, take away from the Holy Spirit what belongs to him. Although by baptism wicked men are neither washed nor renewed, yet it retains that power, as far as relates to God, because, although they reject the grace of God, still it is offered to them. But here Paul addresses believers, in whom baptism is always efficacious, and in whom, therefore, it is properly connected with its truth and efficacy. But by this mode of expression we are reminded that, if we do not wish to annihilate holy baptism, we must prove its efficacy by ‘newness of life.’” (Rom. 6:4)