Short answer: Yes.
Longer answer: Yes, but it’s a little complicated.
Eastern Catholics* have a slightly different system of canon law than Latin (a.k.a. “Roman”) Catholics.** A rough counterpart to our Latin 1983 Code of Canon Law (CIC) is their 1990 Code of Canons of the Eastern Churches. (CCEO)
However, the CCEO is different from the CIC in one very significant way: while the Latin Code of Canon Law governs Latin Catholics in a single flat “layer,” the Code of Canons of the Eastern Churches is written in such a way so as to account for the fact that each individual Eastern Church—or what we would technically call Churches “sui iuris”—also has its own proper law specific to that particular Church. The CCEO sets some basic universal norms for all Eastern Churches, but in numerous places it defers to a sui iuris Church’s proper law.
And as some readers may already know, canon 570 in the Code of Canons of the Eastern Churches explicitly mentions consecrated virgins, along with hermits and consecrated widows. As this canon states:
“Particular law can establish other kinds of ascetics who imitate eremitical life, belonging or not to an institute of consecrated life. Consecrated virgins and widows who live on their own in the world, having publicly professed chastity, can also come under norms of particular law.”
But as we can see here, this is one instance where the actual details are matter of an individual Church’s proper (i.e. “particular”) law. So in contrast with CIC can. 604, which formally recognizes the Ordo virginum as an established form of consecrated life in the Latin Church throughout the world, CCEO can. 570 merely allows for the possibility of individual sui iuris Churches deciding to have the vocation of consecrated virginity within their own ecclesial community.
So the permission to have consecrated virgins in CCEO can. 570 also comes with the implied caveat that an Eastern sui iuris Church could legitimately decide not to have consecrated virgins, or that an Eastern Church could decline to establish the Ordo virginum within their tradition.
And if a sui iuris Eastern Church did decide to have consecrated virgins, there is another issue to be addressed: i.e., how exactly are these women to be consecrated? For Latin Catholics, the Rite of Consecration to a Life of Virginity, in both its modern and historic forms, is part of our own specific and venerable liturgical tradition. That is, the Rite of Consecration that we know and love today is not simply a generic ritual, but has a distinctively “Latin” and “Roman” character. As the 1970 decree promulgating the Rite of Consecration states: “The rite for the consecration of virgins belongs to the treasures of Roman liturgy.”
Because of this, in my opinion it would not be appropriate for an Eastern Church to simply “borrow” the Latin Rite of Consecration to a Life of Virginity when consecrating virgins in their own Eastern ecclesial context. (To do so would be akin to, for example, a Latin priest deciding to celebrate the Byzantine Holy Week liturgies in his Latin parish in lieu of the Latin Triduum services and the Easter Vigil.) In order for an Eastern Church to consecrate a virgin within their own community, the competent authorities would need to either: identify a historic Rite of Consecration that developed as part of their own proper liturgical tradition; or develop a new but characteristically Eastern liturgy for the consecration of virgins “from scratch”; or else somehow combine these two approaches—such as, perhaps, adapting something like an ancient liturgy for the institution of early deaconesses and/or ancient forms of female monastic profession.
In all honesty, I am unfortunately not personally familiar enough with the wide world of Eastern Catholicism to know all the details of which sui iuris Churches are making what provisions to establish their own Order of virgins. But certainly, the establishment of the Ordo virginum in individual Eastern Catholic Churches is a fascinating topic!
Still, in the meantime, what should an Eastern woman do if she feels called to consecrated virginity? The obvious first step would be to reach out to her own Eastern bishop to ask about the possibility of consecration within her own Church.
But if her own Eastern Church does not have provisions for consecrated virginity, one other option—especially in places like the United States, which is predominantly Latin but still has a sizable Eastern representation—would be for the woman to contact her local Latin diocese. If both the woman’s Eastern bishop and the relevant Latin bishop agree, she could be consecrated to a life of virginity by the local Latin bishop, for the local Latin diocese, according to the Latin Rite of Consecration to a Life of Virginity. This would be parallel (cf. CIC can. 17) to scenarios where an Eastern Catholic man is ordained a priest by a Latin bishop and incardinated into a Latin diocese.
Of course, this option is not a “quick fix.” A Eastern woman consecrated in a Latin diocese would need to come to a place in her interior life where she felt she could truly belong to that diocese in a deep spiritual sense. And it could be spiritually, emotionally, and pastorally complicated for some women to essentially have two spiritual homes at the same time—that is, a home within her proper Eastern Church by virtue of her baptism; and a home within a Latin diocese by virtue of her consecration as a virgin. In a case like this, an especially sensitive and careful discernment would be needed on the part of everyone involved.
* For those unfamiliar with the term, Eastern Catholics are Catholics who, while being fully in union with the Pope, worship according to a different liturgical tradition than the “Roman” or Latin Catholics who comprise the majority of the Catholic Church. Often, Eastern Catholicism is connected to a particular geographical area and culture—as just a few examples, Byzantine Catholics are generally of Slavic descent, the Syro-Malabar Church originates in India, an Maronite Church is predominantly Lebanese. Eastern Catholics have their own bishops and are organized into their own dioceses.
** Many people refer to “mainstream” Catholics as “Roman Catholics,” as this largest of sui iuris Churches was founded by St. Peter in Rome, with our liturgical and canon law traditions being broadly influenced by ancient Roman culture. However, often today the preferred term is “Latin Catholic,” as “Roman Catholic” might be seen as downplaying the unity of the Eastern Churches with the Pope, who is Bishop of Rome.