What were the people who didn't agree with the Roman Church called?

Answer 1

Roman Catholic AnswerIf by "Roman Church" you are referring to the Christian Church, commonly known as the Catholic Church, there are many names for those who don't agree with them. Schismatic refers to someone who belongs to a Church that has a valid priesthood, and valid Sacraments, but doesn't acknowledge the Roman Pontiff. Heretic is one who was a Catholic and denies the faith completely. Protestant refers to one who objected to Christian truths and decided they would rather follow their own ideas. Pagans are those who have never been Christian or known the truth, etc. Non-Roman Catholic Answer

A couple of additions/corrections: a heretic is one who follows a heresy (that is, a view other than the official one; it comes from a word meaning "to choose"); it doesn't necessarily have to be denying the faith completely, and in fact many times they boil down to a single difference of opinion, such as the Monophysite heresy, which basically says that Christ had only one "nature" - either divine or a mixture of divine and human - rather than a dual nature, both divine and human. (Yes, it's somewhat pedantic. Many heresies were.) Someone who was a Catholic (or member of any other belief) and no longer is (whether they deny their former beliefs completely, or only portions of them is an apostate, which comes from a word literally meaning "to stand off". Protestant was used for those who protested (see how that works?) against corruption and false doctrine in the leadership of the Roman Catholic church, and broke off from it in order to pursue what they believed to be the true Christian faith... the original protestants were also apostates, but the term today usually means "someone who follows the tenets of one of the original protestants", or even simply as a catchall for "Christian other than Roman Catholic or Eastern Orthodox."

In any case, I think the word you're looking for is probably "Protestant."

Answer 2

Roman Catholic AnswerProbably a heretic, it depends on who is doing the criticizing. Below are a list of words and their meanings that would seem to fit with your question.

from

The Catechism of the Catholic Church, second edition, English translation 1994

2089

Incredulity is the neglect of revealed truth or the willful refusal to assent to it. "Heresy is the obstinate post-baptismal denial of some truth which must be believed with divine and catholic faith, or it is likewise an obstinate doubt concerning the same; apostasy is the total repudiation of the Christian faith; schism is the refusal of submission to the Roman Pontiff or of communion with the members of the Church subject to him." (Codex Iuris Canonici, canon 751: emphasis added.)

from

Modern Catholic Dictionary by John A. Hardon, S.J. Doubleday & Co., Inc. Garden City, NY 1980

Heresy. Commonly refers to a doctrinal belief held in opposition to the recognized standards of an establish system of thought. Theologically it means an opinion at variance with the authorized teachings of any church, notably the Christian, and especially when this promotes separation from the main body of faithful believers.

In the Roman Catholic Church, heresy has a very specific meaning. Anyone who, after receiving baptism, while remaining nominally a Christian, pertinaciously denies or doubts any of the truths that must be believed with divine and Catholic faith is considered a heretic. Accordingly four elements must be verified to constitute formal heresy; previous valid baptism, which need not have been in the Catholic Church; external profession of still being a Christian, otherwise a person becomes an apostate; outright denial or positive doubt regarding a truth that the Catholic Church has actually proposed as revealed by God; and the disbelief must be morally culpable, where a nominal Christian refuses to accept what he knows is a doctrinal imperative.

Objectively, therefore, to become a heretic in the strict canonical sense and be excommunicated from the faithful, one must deny or question a truth that is taught not merely on the authority of the Church but on the word of God revealed in the Scriptures or sacred tradition. Subjectively a person must recognize his obligation to believe. If he acts in good faith, as with most persons brought up in non-Catholic surroundings, the heresy is only material and implies neither guilt nor sin against faith. (Etym. Latin haeresis, from the Greek hairesis, a taking, choice, sect, heresy.)