Women cannot be ordained to the priesthood.
For Catholics this is a settled issue: women cannot properly be ordained to the ministerial priesthood.
Fr. Jordi Rivero

St. John Paul II formally declared that the Church does not have the power to ordain women and, in 1995, the Congregation for the Doctrine of the Faith, in conjunction with the pope, ruled that 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. >>

Church documents: Inter insigniores and Ordinatio Sacerdotalis.

Inter Insegniores Inter-Insegniores, Declaration on the Admission of Women to the Ministerial Priesthood (15 October 1976), Sacred Congregation for the Doctrine of the Faith Ban on Women Priests is Infallible

For more study on this topic read: The Catholic Priesthood and Women: A Guide to the Teaching of the Catholic Church by Sister Sara Butler, professor of dogmatic theology. She is a former advocate of women ordination.

Her theses is that one can view the Catholic priesthood either socially or sacramentally. A social view sees the priesthood merely as an office of leadership or a career. But the priesthood is a sacrament. The ordained ministers serve as "signs" or icons of Christ. Christ chose for the priesthood "those whom He wanted" (Mark 3:13). Male priesthood has been the unbroken and universal tradition. The Church must be obedient to Christ and therefore has no authority to change the priesthood by ordaining women. 

Women do not have a "social right" to ordination. Ordination is not a issue of civil rights since no one has the right to be ordained. Priesthood cannot be understood without first understanding what is a sacrament, as a reality of divine, not human origin. 

John Paul II and Benedict XVI have made it clear that not ordaining women does not imply a negative judgment on women. The Church regards men and women as equals in Christ (cf. Gal 3:28) and rejects the idea that the equalization of rights requires the identical treatment of women and men".

Common objections to Church teaching:

The notion that Jesus chose no women to be Apostles because of the culture in which He lived. But Jesus never compromised the Truth by conforming to societal constraints nor intimidation.  Nor should the Church.

That the Church is oppressing women. But the 1983 Code of Canon Law is clear that Catholic women have essentially the same juridical status as Catholic men