Reformed theology includes covenant theology, whereby the nation of Israel is understood to be the church of the Old Testament. In this light, there is no distinction between an Israelite Psalm and a Christian Psalm; they are one and the same.
Not only this, but the Reformers consistently sang almost only the 150 Psalms from the Book of Psalms (as well as the Songs of Simeon, Zachariah, and Mary, the Apostle's Creed, the Ten Commandments, the Lord's Prayer, and a very small number of other hymns).1.2.3.4.
John Calvin, who was the single largest influence on the Reformed understanding on how to worship, said in his Letter to the Reader introducing the Genevan Psalter the following:
But what St. Augustine says is true, that no one can sing things worthy
of God unless he has received them from him. For when we have searched
here and there, we will not find better songs nor ones more appropriate
for this purpose than the Psalms of David, which the Holy Spirit has
spoken to him and made. Therefore, when we sing them, we are certain
that God has put the words in our mouth as if they themselves sang in us
to exalt his glory. Consequently Chrysostom exhorts both men, women
and little children to learn to sing them in order that they may be like a
meditation to associate them with the company of angels (Chrysostom,
In Ps. 41.1,2).
Calvin was in the habit of preaching from either the New Testament or the Book of Psalms every Sunday afternoon, and it is from these sermons on the Psalms that we get his commentary on the Psalms.
In addition, the Westminster Confession of Faith, Chapter 21 Article 5 says the following (emphasis mine):
The reading of the Scriptures with godly fear, the sound preaching and conscionable hearing of the Word, in obedience unto God, with understanding, faith and reverence, singing of psalms with grace in the heart; as also, the due administration and worthy receiving of the sacraments instituted by Christ, are all parts of the ordinary religious worship of God: beside religious oaths, vows, solemn fastings, and thanksgivings upon special occasions, which are, in their several times and seasons, to be used in an holy and religious manner.
While this is not to mean the exclusion of non-Book of Psalms songs, certainly the creators and users of the Westminster Standards (a classic Reformed summary of what the Bible teaches) primarily sang Psalms and considered them to be appropriate for the Christian church.
I will also note that it is not completely precise to refer to the Book of Psalms as a Jewish book; it is an Israelite book, from before the splitting of the nation of Israel into two kingdoms.
