On the contrary, the word "church" does appear in the Scriptures, and in a meaning completely congruent with the idea of "the church universal".
Initially, it appears that congregations of "followers of the Way" - as Christians were called early on - gathered in individual houses; it wasn't until perhaps the early third century that buildings began to be set aside for Christian worship. But even in the mid-first century a regular gathering of Christians was known as a "church" - an ekklesia in Greek. Paul uses the word in his epistles:
Paul, Silas and Timothy,
To the church of the Thessalonians in God the Father and the Lord Jesus Christ
(1 Thessalonians 1:1, NIV)
Paul, an apostle... to the churches in Galatia
(Galatians 1:1–2, NIV)
In both of these cases (there are more), the word used is an inflection of the Greek ekklesia.
The word "church", then, is indeed Scriptural. And the term "Church Universal" (ekklesia katholike)? The specific phrase does not appear in the New Testament, but the concept of "the Church spread throughout the world" does.
Acts 8:1–3 describes the first persecution of the Church:
On that day a great persecution broke out against the church in Jerusalem, and all except the apostles were scattered throughout Judea and Samaria. Godly men buried Stephen and mourned deeply for him. But Saul began to destroy the church.
Here is a clear distinction between "the church in Jerusalem", a local community of believers, and "the church" in general. (It is clear that it was not simply the church in Jerusalem that Saul intended to destroy; in chapter 9 he sets out to do the same in Damascus.)
More strongly, Matthew 16:18 has Jesus say: "'I tell you that you are Peter, and on this rock I will build my church." Clearly, this means "the whole community of believers throughout the world" - the meaning of "the Church Universal".
Finally, Revelation begins with Paul's salutation:
John, to the seven churches in the province of Asia:
(Revelation 1:4, NIV)
With regard to this phrase, Ellicott's Commentary for English Readers states:
The churches are types and representatives of the whole family of God. Every community may find its likeness here. This much is admitted by the best commentators of all schools. “The seven churches,” says St. Chrysostom, “are all churches by reason of the seven Spirits.” “By the seven,” writes St. Augustine, “is signified the perfection of the Church universal, and by writing to the seven he shows the fulness of one.”
This is exactly what is conveyed by the phrase "Church Universal" - the community of all followers of Christ, wherever they live and worship.