A Catholic priest must take a vow of celibacy (i.e. unmarried) and thus expected to remain continent (i.e. abstain from sex) so as not to commit a fornication. Is a priest permitted to adopt a child, given that it does not break this vow? Would it matter if they were the only surviving relative of the child?
Yes, a Roman Catholic Priest can adopt children. In 1981, the Vatican granted permission to Fr. George Clements, of Holy Angels Catholic Church in Chicago the permission to adopt a child, and he later adopted three more. I expect another Catholic Priest seeking to do so would need permission, but not necessarily from the Vatican.
The short answer is, “it is theoretically possible but unlikely and uncommon.”
I will begin by pointing out that some Catholic priests are, in fact, married: most of these are Eastern Catholic clergy or converts from Anglicanism who were married before being ordained priests; the rest are priests who have been laicized (i.e., they are forbidden from exercising any ministry and, canonically, are reduced to the lay state) and have received an indult releasing them from their obligation to celibacy.
In these cases, adopting a child poses no particular difficulty.
Assuming we’re talking about a priest from the Western Church still in good standing (i.e., not laicized), Canon Law does not prohibit adoption—indeed it does not mention it at all.
That being said, a celibate priest adopting a child poses a number of difficulties:
From the moral perspective, the child would be deprived of a motherly figure. The adopting priest would effectively be a single parent. Unless there is an urgent need (e.g., the child has no one else to care for him), a child should always be adopted by a two-parent family (two parents of the opposite sex, evidently).
Legal adoption imposes a number of burdens on the adopter. Since priests are always incardinated in (i.e., are members of) a diocese, religious congregation, or similar organization, these obligations could eventually have repercussions for the diocese. (See Code of Canon Law, Can. 266.)
Adopting a child would impose obligations on the priest that would take away from his pastoral duties.
There is the possibility of scandal: if the faithful or others see a priest taking care of a child, they could easily come to the (erroneous) conclusion that he has broken his obligation of celibacy.
One could, however, conceive of cases in which adoption is a possibility: in particular, in cases in which a given child has no one else to take care of him.
If a case like this were to arise, because of the obligations placed on the diocese or institute the priest belongs to and the possibility of scandal, the priest would have to obtain permission from his bishop or religious superior before legally adopting a child. (Simply taking care of a child while he is looking for a couple to adopt him would be a different matter—especially in parts of the world with no foster care or similar program in place.)
Evidently, the Pope can also grant that kind of permission.
However, it would be exceedingly rare for a diocese or institute to grant such a permission, because in the vast majority of cases, it is far better (and even more feasible) to find a couple willing to adopt. Only in a truly exceptional case would it be a good idea for a priest to do so.
Assuming civil law allows the clergy to adopt, an answer would have to address whether Catholic canon law permits it. It would seem not because fathering children would detract from his obligations to the Church (which is a reason why priests cannot marry); he certainly could run an orphanage for such children.
If the Pope could, then a priest could.
It is commonly believed that Pope Pius IX "adopted" a boy in very extenuating circumstances in 1856 and created in international scandal.
The Mortara Affair began on this date in 1856 when a contingent of papal police kidnapped 6-year-old Edgardo Mortara from his parent’s apartment in Bologna because Church higher-ups had learned that the child had been secretly baptized by a maid five years earlier when he was seriously ill. Edgardo was whisked to Rome, “adopted” by Pope Pius IX, and eventually became a priest. Jewish organizations and international leaders protested to the Vatican and called for Edgardo to be returned to his parents, but the pope declared that canon law forbade non-Christians in the Papal States to raise a Christian child, even if the child was their own. Protests in France led to the founding of the Alliance Israelite Universelle in 1860, and helped fuel the movement for Italian unification, which overcame the Papal States in 1870. As an adult, Edgardo was sent on missionizing missions to Jews in Germany and the United States, and vigorously sought the beatification of Pope Pius IX — which finally took place in 2000. Mortara did reconnect to his parents and attended his mother’s funeral in Bologna in 1895. He himself died at 88 in 1940. - June 24: The Mortara Affair
Here is what Wikipedia has to say of the Moratara Case.
Pope Pius IX's determination to keep Edgardo developed into a strong paternal attachment. According to Edgardo's memoirs, the pontiff regularly spent time with him and played with him; the Pope would amuse the child by hiding him under his cassock and calling out: "Where's the boy?" At one of their meetings, Pope Pius told Edgardo: "My son, you have cost me dearly, and I have suffered a great deal because of you." He then said to others present: "Both the powerful and the powerless tried to steal this boy from me, and accused me of being barbarous and pitiless. They cried for his parents, but they failed to recognize that I, too, am his father.
As Brasshat points out in his answer that it is possible for a priest to adopt a Child. I know of some American nuns (sisters) who have receive permission to adopt children.
Blessed Charles de Foucauld (1858-1915) began a program of buying slaves in 1902 in order to free them. For Blessed Charles this was done as a form to encourage adoption. In addition to being adopted himself, he is also one of the Patron Saints of the Adopted in some Catholic circles.
Our Father never failed to add: «Someone will say to me: “But what about me, I have no family to speak of, I have no traditions, I am one of those serfs produced by modern society; so what is to become of me?” I shall answer: “Well, allow yourself to be adopted by Father de Foucauld!” Here we have a family which is open to us through the generosity of this heart of gold. Just as he loved all the members of his own family, he would have desired to have brothers and sisters, and even an entire people of lay folk who would be attached to his work, that is to say, who would share his love for neighbour and then for those furthest away. So, if we have no family, let us allow ourselves to be inserted into this great family of age old France and of the Catholic Church!» - BLESSED CHARLES DE FOUCAULD