Any massive change in the historical conditions will affect all aspects of culture and society.
Make no mistake: an Industrial Revolution in the 16th century is a massive historical change. An industrial revolution requires a large, educated, literate, and free workforce; it requires an active economy; it requires universal education; it requires an efficient transportation network; it requires an effective banking system, extending credit to those entrepreneurs who build all those machines and factories; in a word, it requires a world vastly different from the world of the late Middle Ages and the Renaissance.
And in a vastly different world you will have a different Church.
In particular, the evolution of the doctrines and discipline of the Catholic Church will of course be changed if 15th century Europe goes on the path of Industrial Revolution. How specifically will the evolution of the doctrines and discipline of the Catholic Church be affected we don't know, of course, and exploring this change could be a very interesting aspect of the story to be invented by the author.
In real history, there actually was a massive change in the 15th century which left its mark on the evolution of the doctrines and discipline of the Catholic Church, namely, the Reformation.
The Reformation movement triggered a "period of Catholic resurgence" (the words are from Wikipedia) generally known as the Counter-Reformation. A high point of this fervent spiritual activity was the Council of Trent (1545-1563); among others, the Council of Trent:
Established that the deuterocanonical books "were on a par with the other books of the canon" (again words from Wikipedia);
Affirmed that human action (specifically, committing a mortal sin) can indeed forfeit the grace of God;
Reaffirmed that the ordination of priests is a Holy Sacrament and it confers an indelible character on the soul.
One of the notable effects of the Council of Trent was the compilation of an authoritative version of the Bible, commonly known as the Vulgate.
Sacrosancta Oecumenica, & generalis Tridentina Synodus, in Spiritu Sancto legitime congregata, [...] proponens, ut, sublatis erroribus, puritas ipsa Evangelii in Ecclesia conservetur.
(Fun factoid: the first authoritative version of the Catholic Bible was published in 1590, almost 60 years after the publication of Luther's German Bible, which was printed in 1534.)
And the dogmas and discipline of the Catholic Church continued to evolve after the 16th century, responding to changes in the world at large. In 1854, the Catholic Church proceeded to officialize the dogma of the immaculate conception of the Mother of God; in 1870, the First Vatican Council proclaimed the dogma of papal infallibility; the policy of encouraging the emergence of the Eastern Catholic Churches (from the 17th century onwards) required some flexibility in the intepretation of the dogmas and discipline of the Latin Church (which, in America and western Europe, remains the best known of the 24 autonomous churches which make up the Catholic Church).
(Fun factoid: while the Latin Church enforces strict clerical celibacy, most of the Eastern Catholic Churches allow the ordination of married men. The point being that there are very many Catholic priests who are happily married; they do not belong to the Latin Church, but that doesn't make them any less Catholic.)
All in all, exploring how the doctrines and discipline of the Catholic Church would respond to 16th century Industrial Revolution would be a very interesting element of the story. I would be quite interested to read a learned essay on the topic.