All the quotes below are taken from Monsignor Joseph Fenton’s article titled “Contemporary Questions about Membership in the Church” published in the July 1961 Edition of The American Ecclesiastical Review and provided by Catholic Culture. There are paragraphs in the article that are not included in between the quotes. This is indicated by the term “BREAK“. The bold emphases are mine.
“Certainly the Mystici Corporis Christi statement about membership in the Church is quite in line with the teaching of the De ecclesia militante. According to Pope Pius XII, four factors alone are necessary in order that a man be counted as a member of the true Church. These are (1) the reception of baptism, and thus the possession of the baptismal character, (2) the profession of the true faith, which is, of course, the faith of the Catholic Church, (3) the fact that a person has not cut himself away from the structure or the fabric of the ‘Body,’ which is, of course, the Church itself, and (4) the fact that a person has not been expelled from the membership of the Church by competent ecclesiastical authority.
“It is the nature of the third of these four factors which, in the context of the encyclical, is not completely clear. Very definitely a person would cut himself off from the structure of the ecclesiastical Body if he entered into a state of public heresy or apostasy. But that condition had already been taken care of in the naming of the second of the factors which the Mystici Corporis Christi lists as requisite for membership in the true Church. Very definitely the ‘cutting away’ mentioned in the third point of this statement might involve entrance into the state of schism. But it could, of course, imply that some act against the spiritual or invisible bond of unity within the Church might also cut a person away from membership in the Church. The text of the Mystici Corporis Christi is not, in itself, sufficiently clear on this point.
“Yet, over the course of the years, it has become increasingly obvious that the common teaching of the Catholic theologians holds that people are members of the Church or parts of the Church only by the possession of these visible or palpable factors. The term ‘member of the Church’ can legitimately be applied only to those baptized persons who have not frustrated the force of their baptismal characters by public heresy or apostasy, or by schism, and who have not been expelled from the Church by competent ecclesiastical authority. The theological demonstration that backs up this thesis is still and always will be the ‘proof from reason’ which St. Robert Bellarmine alleged in support of his teaching in the De ecclesia militante.11 More effectively, perhaps, than any other writer in the history of the Catholic Church, St. Robert pointed to the fact that the basic Catholic claim, that the Church militant according to the dispensation of the New Testament is essentially a visible Church, involves and includes the teaching that membership in the Church is possessed by all and only the people who have those factors which go to make up the visible or external bond of unity within the Church of God.”
“(4) The baptismal character is the basic force incorporating a man as a member into the true Church of Jesus Christ in this world according to the dispensation of the New Testament. Yet it is quite obvious that not every baptized person is a Catholic. Very definitely the society which is the one and only supernatural kingdom of God in this world is not made up or composed of all baptized persons. The unifying force of the baptismal character can be and is frustrated by public heresy or apostasy, by schism, and by expulsion from the Church. To say or even to insinuate that all baptized people are members of the Church is to deny, at least by implication, the central dogma of ecclesiology, the divinely revealed teaching that tells us that the Roman Catholic Church, the religious society which recognizes and accepts the Bishop of Rome as its visible head, is actually the Mystical Body of Jesus Christ.”
These quotes show that public heresy is incompatible with being a Catholic.