The question was considered by the International Theological Commission (dependant of the Vatican department known as the Congregation of Faith) in a 2002 document called From the Diakonia of Christ to the Diakonia of the Apostles. After revising the theological and historical background of the question, the commission concludes that:

"it pertains to the ministry of discernment which the Lord established in his Church to pronounce authoritatively on this question".

Meaning that there is no sufficient theological, historical or dogmatic reason why this cannot be, but depends exclusively on what the Church decides. The question is still open and just depends on the willingness of the Church leaders. In October of 2015, during the Synod of the Family, the Canadian Archbishop Paul-Andre Durocher of Gatineau, Quebec raised the question once again. No official answer was given.

The reason that the Church does not ordain women as deacons is similar to that for not ordaining women as priests. (There is a separate question answering that.)

In essence, the answer is that the Church only has the power to act with those powers and abilities that Jesus has entrusted to it; it has no capacity do otherwise.

For example, Jesus gave priests and bishops the power to confect the Eucharist: that is, to convert wheat bread and grape wine into the Substance of Jesus Christ. He did not give them power, however, to confect the Eucharist using other substances, and attempting to do so would be without effect. It is impossible, for example, to confect the Eucharist using rice cakes, or wine made from fruit other than grape. If there were a shortage of either wheat or grapes, the Church would simply have to refrain from celebrating the Eucharist, since there is no other “matter” available. (See a document from the Sacred Congregation for the Sacraments and Divine Worship called Inaestimabile donum, no. 8.)

In a similar way, Jesus gave bishops the power to ordain baptized males to the diaconate, the priesthood, and to the episcopate. He did not give them the ability to ordain women (certainly not to the priesthood or to the episcopate—see below), and so no matter how pressing is the situation, ordaining women is simply not possible. (In fact, attempting to do so would be without effect.)

The only difference, as regards the possibility of ordaining women, between the diaconate and the presbyterate (priesthood) is that the Magisterium (the teaching authority) of the Church has pronounced definitively on the impossibility of ordaining women to the priesthood. Pope John Paul II issued an apostolic letter called Ordinatio sacerdotalis in 1994, which says,

Wherefore, 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 (No. 4).

Although ordination to the diaconate is not expressly excluded in this document, it is nearly certain that the exclusion also applies to that case, because the Sacrament of Holy Orders is a unified whole, albeit with three degrees, and also because it has been the universal practice of the Church to ordain only males, even to the diaconate, from the beginning.

Deaconesses?

At this point, attentive readers of the New Testament will note that in a few cases, women are called diakonoi, which is the term that came to be used for members of the first degree of holy orders (i.e., deacons). For example, there is Romans 16:1:

I commend to you our sister Phoebe, a servant [diakonon] of the church at Cenchreae (ESV).

The problem is that, as the ESV suggests, the term “diakonos” was much broader in meaning in Koine Greek than it is today: it essentially meant “servant.” (For example the servants who fill the water jugs at the wedding at Cana, in John 2:5, are also called diakonoi.)

There is evidence that the early Church had orders of “deaconesses” who would, for example, assist adult women in their full-immersion Baptisms (which was the norm back then). It is clear by all accounts, however, that the deaconesses never received the Sacrament of Holy Orders through the imposition of hands, as (male) deacons do. Again, the term “deaconess” was applied before the term “deacon” came to have the technical use it has today. (There is a good summary of this issue in the old Catholic Encyclopedia).