The Church has considered the question of the priestly ordination of women for nearly 40 years. The more recent, and more authoritative, pronouncement on the matter is in Ordinatio Sacerdotalis, an Apostolic Letter of Pope St. John Paul II, published in 1994. In this letter, the Pope concluded that
in order that all doubt may be removed regarding a matter of great importance, a matter which pertains to the Church's divine constitution itself, in virtue of my ministry of confirming the brethren (cf. Lk 22:32) I declare that the Church has no authority whatsoever to confer priestly ordination on women and that this judgment is to be definitively held by all the Church's faithful.
Shortly after this letter was promulgated, the question was raised as to whether it was to be held as infallible, or at least as belonging to the Church's depositum fidei, the "deposit of faith" containing truths to be believed by all Catholics. In 1995, shortly after this question was put to the Congregation for the Doctrine of the Faith, they answered in the affirmative:
This teaching requires definitive assent, since, founded on the written Word of God, and from the beginning constantly preserved and applied in the Tradition of the Church, it has been set forth infallibly by the ordinary and universal Magisterium (cf. Second Vatican Council, Dogmatic Constitution on the Church Lumen Gentium 25, 2). Thus, in the present circumstances, the Roman Pontiff, exercising his proper office of confirming the brethren (cf. Lk 22:32), has handed on this same teaching by a formal declaration, explicitly stating what is to be held always, everywhere, and by all, as belonging to the deposit of the faith.
But the older discussion of the question, referred to in John Paul II's letter, appears in Inter Insigniores, a Declaration of the Congregation of the Faith issued in 1976. This letter considered the (at that time) relatively new great increase in the sphere of women's activity:
Our age gives rise to increased demands: "Since in our time women have an ever more active share in the whole life of society, it is very important that they participate more widely also in the various sectors of the Church's apostolate". ... A number of Catholic theologians have even posed this question [of whether women may be ordained priests] publicly, evoking studies not only in the sphere of exegesis, patrology and Church history but also in the field of the history of institutions and customs, of sociology and of psychology. The various arguments capable of clarifying this important problem have been submitted to a critical examination. As we are dealing with a debate which classical theology scarcely touched upon, the current argumentation runs the risk of neglecting essential elements.
(Inter Insigniores, Introduction. The quote is from Apostolicam Actuositatem, a decree of the Second Vatican Council.)
This letter, like the later one of Pope John Paul II, concluded that
the Sacred Congregation for the Doctrine of the Faith judges it necessary to recall that the Church, in fidelity to the example of the Lord, does not consider herself authorized to admit women to priestly ordination.
With respect specifically to the claim that a woman may feel herself called to the priesthood, the Congregation stated that
It is sometimes said and written in books and periodicals that some women feel that they have a vocation to the priesthood. Such an attraction however noble and understandable, still does not suffice for a genuine vocation. In fact a vocation cannot be reduced to a mere personal attraction, which can remain purely subjective. Since the priesthood is a particular ministry of which the Church has received the charge and the control, authentication by the Church is indispensable here and is a constitutive part of the vocation: Christ chose "those he wanted" (Mk.3:13). On the other hand, there is a universal vocation of all the baptized to the exercise of the royal priesthood by offering their lives to God and by giving witness for his praise.
Women who express a desire for the ministerial priesthood are doubtless motivated by the desire to serve Christ and the Church. And it is not surprising that, at a time when they are becoming more aware of the discriminations to which they have been subjected, they should desire the ministerial priesthood itself. But it must not be forgotten that the priesthood does not form part of the rights of the individual, but stems from the economy of the mystery of Christ and the Church. The priestly office cannot become the goal of social advancement: no merely human progress of society or of the individual can of itself give access to it: it is of another order.
In other words, if women feel they are being called to the priesthood, the Church believes, they are mistaken: the ministerial priesthood (that is, the office of offering sacrifice and assisting the Bishop in caring for the flock of Christ) is not an office open to them. Admission to the priesthood is not a right of any individual; it is a calling of God which requires the formal assent of the Church. Women are called, as all are, to the "royal priesthood" (cf. 1 Peter 2:9, Isaiah 61:6) of all believers, but the Church has no authority to admit them to the ministry of priests.