What is the theological justification for only the priest drinking the wine in a Catholic Mass? I recently attended a Catholic Mass for a funeral.  When it came time for the observance of the Lord's Supper, only the priest drank the wine.  There may be some protestant churches which also follow this practice, but any church I've attended always handed out both the bread and the wine to all of the partakers.  

It would seem, from a natural reading of the Biblical account, that both the bread and the wine were intended to be part of the observance.  So, my question is, what is the basis, scriptural or otherwise, for only the priest needing to drink the wine?

I found this other question, which was helpful in understanding the logistical reasons for not wanting to have to do both.  It also explained how either is optional, as long as you take one.  But it failed to answer my question regarding the theological reasoning and explanation.  Does anyone know when this practice first became accepted?  What is the theological basis for only one or the other (body and blood) being necessary?
Communion under Both Kinds article at Catholic Encyclopedia has answer to both of your questions

Does anyone know when this practice first became accepted?

During early times public Communion in the churches was received under both kinds. But side by side with the regular liturgical usage of Communion, there existed from the earliest times the custom of communicating in certain cases under one kind alone. Examples of receiving communion in one kind alone can be seen as early as 3rd century in the writings of Tertullian, St. Cyprian, Eusebius and others. In these writing we can get an idea of why this practice might have been started. Some primary reasons are:


That Communion of the sick is easier to transport and administer under the species of bread alone
In the Early Church the Holy Eucharist in the species of wine alone was given to children even before they attained the age of reason. Probably because it is easy to swallow and risk of children spitting it out is considerably reduced
During the Mass of the Presanctified and all fast days during Lent, the faithful received Communion only under the species of bread.


What is the theological basis for only one or the other (body and blood) being necessary?

Catholic Church does not say it is necessary to receive Communion in one species alone. It says it is sufficient to receive Communion in one species alone.


  There is no Divine precept binding the laity or non-celebrating priests to receive the sacrament under both kinds (Trent, sess. XXI, c. i.) 
  
  By reason of the hypostatic union and of the indivisibility of His glorified humanity, Christ is really present and is received whole and entire, body and blood, soul and Divinity, under either species alone; nor, as regards the fruits of the sacrament, is the communicant under one kind deprived of any grace necessary for salvation (Trent, Sess. XXI, c., iii).


1 Corinthians 11 : 27-29 states 'if anyone eats this bread or drinks this cup of the Lord unworthily, he will be held to account for the Lord’s body and blood.' Here St. Paul attaches the same guilt "of the body and the blood of the Lord" (copulative) to the unworthy "eating or drinking"(disjunctive). In its article The Real Presence of Christ in the Eucharist, Catholic Encyclopedia points out that based on this verse Catholic church believes that Jesus, in the species of bread alone, is fully present with his entire body, blood, soul and Divinity. 

Hence as far as the effects are concerned, those who receive under only one species are not deprived of any of the grace that is necessary for salvation.

The main reason for the Catholic faithful not to partake in the cup is because of practical purposes. It is quite impossible to provide the cup for hundreds of people in a mass and protect the sacred species from spilling to ground. This is the only reason for which currently faithful are not required to receive in both forms on any given day.

As pointed out by @curiousdannii and @Andrew Leach, Theologically speaking preventing the sacred species from spilling to ground is a big deal because of what the Church believes it it be. That is Christ himself. So spilling the species (desecration) is among the gravest of sins. (Such actions, if done willfully, can be forgiven only be the Pope on the recommendation of the concern Roman curia. This shows the gravity of the matter in the Catholic Church) Hence it must be avoided whenever possible.

You might now ask: When can the faithful receive Communion under both kinds? The General Instruction of the Roman Missal, 283 and Eucharisticum Mysterium - 32 points out that apart from the priests who celebrate/concelebrate the mass following can receive communion in both species :


  
  To newly baptized adults in the Mass which follows their baptism; to confirmed adults in the Mass of their Confirmation; to baptized persons who are received into communion with the Church;
  To bride and bridegroom in the Mass of their wedding;
  To newly ordained in the Mass of their ordination;
  To abbesses in the Mass of their blessing; to virgins in the Mass of their consecration; to professed in the Mass of their first or renewed religious profession, provided that they take or renew their vows during the Mass;
  To lay missionaries, in the Mass in which they are publicly sent out on their mission, and to all others in the Mass in which they receive an ecclesiastical mission;
  In the administration of Viaticum, to the sick person and to all who are present when Mass is celebrated in the house of the sick person, in accordance with the existing norms;
  To deacon, subdeacon, and ministers, who carry out their ministry in a solemn or pontifical Mass;
  When there is a concelebration:
  
  a) to all who exercise a genuine liturgical function in this concelebration, including lay people and to all seminarians who are present;
  
  b) in their churches, to all members of institutes practicing the evangelical virtues and of other societies in which the members either through religious vows or offering or a promise dedicate themselves to God; and also to all those who normally live in the house of the members of these institutes and societies.
  To priests who take part in large celebrations, but are not able to celebrate or concelebrate;
  To all groups which are making retreats or following spiritual exercises, in a Mass which is celebrated during the retreat or exercises for those who are taking part; to all those who are taking part in the meeting of some pastoral commission, in the Mass they celebrate in common;
  To those listed under nos. 2 and 4, in their jubilee Masses;
  To the godfather, godmother, parents and spouse of a baptized adult, together with the lay catechists who have prepared him, in the Mass of initiation;
  To the parents, relatives and special benefactors, who take part in the Mass of a newly ordained priest.
  


But as with most disciplinary regulations of the Church, the diocesan Bishop may establish norms for Communion under both kinds for his own diocese permitting the parish priest to decide on whether to administer the sacrament in both kind (GIRM 283), as long as there is no danger of profanation of the Sacrament or of the rite’s becoming difficult because of the large number of participants or some other reason.