Most Protestants do not seem very distressed over the multiplicity of denominations because, in their minds, the Church of Christ is to be found invisibly scattered among all true believers. Technically speaking, we can say that Protestant ecclesiology does not hold the view that the Church is in any sense a visible organism. (This is from a book entitled His Broken Body by Laurent Cleenewerck, published in 2007 by Euclid University Consortium Press [Washington, D.C.], p. 32.)Unlike the denominationalism of the Protestant world, the various churches of Orthodoxy really do have to talk to each other and work things out. A Presbyterian and a Lutheran may each recognize each other as Christian, but they have almost no stake in each other’s internal church life. The same even holds true of someone belonging to the PCA and someone belonging to the PCUSA (both Presbyterian denominations). They don’t have to work anything out between them. A PCA church plant does not in any way infringe on the territory of the PCUSA, because they’re not the same church.
Orthodoxy may often bicker and fight (though most parishioners never see this unless they happen to be in a dysfunctional parish), but the fact that we have such bickering and fighting with each other means that we recognize in each other that we are one Church, that we have a problem and that we need to fix it. Protestants always have the option of just splitting (and once splits occur, they don’t have to bother with each other), while Roman Catholics can ultimately appeal to the Vatican, who can impose solutions that work for the Vatican but might not work for everyone else involved. (This is from "12 Reasons Why I Became and/or Remain an Orthodox Christian," by Father Andrew Stephen Damick, from his blog "Roads from Emmaus," reason #5: "Orthodoxy really is one Church.")
The problem with this is that Protestants, or at least historic Reformed Protestants, do actually have the concept of the catholic church as a visible organic body, in spite of our failure oftentimes to fully realize it and put it into practice. (We can learn something from the Romanists and the Greeks on this point.)
Two historic Reformed writings on church government (among many others--I just happened to be reading in both of these recently) bring this point out clearly: Samuel Rutherford's Due Right of Presbyteries (1644), and Samuel Hudson's A Vindication of the Essence and Unity of the Church-Catholick Visible (1658). These are both excellent books that I would highly recommend. Rutherford's book is extremely long and can be tough reading. Hudson's book is a bit shorter and somewhat easier to read, with a more refined central thesis. Below are a few quotations from each of these.
First of all, here is a quotation from Hudson's "The Epistle Dedicatory," addressed to "the Reverend Assembly of Divines assembled at Westminster":
My principal scope in this and the former Thesis, is to prove that there is one Church-Catholick visible on earth, and that God's intention and donation of the Ordinances of worship and discipline, was first to the whole Church, and secondarily to the particular Churches, as parts thereof. And yet I acknowledge the ordinary and constant exercise of those Ordinances is primarily in the particular Churches, and a secondary and onely occasional exercise of them in greater parts thereof; and a very rare exercise of them in the whole conjunctim upon some general extraordinary occasion, and that can be no otherwise, then by delegated commissioners from the several parts of the whole, when convenible.
Edmund Callamy wrote an "Epistle to the Reader" at the front of Hudson's book. Here is a selection from it:
For the truth is, the position there held forth, if granted, would utterly overthrow the grounds and pillars of the Congregationall government. For if there be a Church-Catholick visible, and this Church be not onely a Church-Entitive but a Church Organicall, and a Totum integrale having all Church-power habitually seated in the Officers of it, which they have commission from Christ to exert, and put into act upon a lawfull call. And if particular Congregations are integrall parts and members of the Church-Catholick, as the Jewish Synagogues were of the Jewish Church. And if the Ministry, Ordinances, and censures were given by Christ first to the Church-general visible, and secondarily to the Church particular, Then it will necessarily follow, That the particular Congregation is not the first receptacle of Church-power, And that all Church-power is not intirely and independently in a particular Congregation, which are two of the chief foundations of the Congregationall government. I shall not at all speak to the first, but as for this last, That all Church-power is solely and independently in a particular Congregation, it seems to me not onely to be contrary to the Scriptures, but to the very light of nature, and to carry many great absurdities with it. For,
1. It takes away all authoritative appeals, and all authoritative waies of uniting particular Churches one with another.
2. Then the Churches of Jesus Christ should have no Church-communion in discipline one with another. They may have Christian-communion, but no Church-communion.
3. Then no Minister could preach as an Officer out of his own Congregation, but onely as a gifted brother, and as a private Christian.
4. Then no Minister could administer the Sacraments (which is an act of office) out of his own Congregation, nor (as I conceive) give the Sacrament to a member of another Congregation.
5. Then when his particular Church is dissolved, hee ceaseth to be a Minister, and must receive a New Ordination.
6. Then a Minister baptizing a childe, baptizeth him onely into his own Congregation. For if he be not an Officer of the Catholick Church, he cannot baptize into the Catholick Church, which is directly contrary to 1 Cor. 12.13.
7. Then when the Officers excommunicate a person, he should onely be excommunicated out of that particular Congregation, &c.
8. Then Christ should have as many intire bodies as particular Congregations; Christ should not onely have one Body whereof particular Congregations are part, but every Congregation should be a Body of Christ by it selfe.
9. It would make way for toleration of heresies and blasphemies, and let in as many religions as there are particular Congregations.
10. It would make the Churches of Christ stand divided one from another in respect of government, and thereby bring ruine upon one another. Even as in a civill state, if particular Corporations should be independent from the whole in point of government, it would quickly bring destruction upon the whole.
For the removing of these and such like absurdities, This learned learned and judicious Author [speaking of Samuel Hudson] in the Book fore-mentioned laid down a quite contrary Thesis. That there is a Catholick visible, organicall Church, to which Ordinances and censures are firstly given by Jesus Christ. And that every Minister is seated by God in this Catholick visible Church, and hath a virtuall and habitual power to preach as a Minister in any place where he shall be lawfully called. Indeed he is not an actuall Minister of the Church-Catholick, nor hath actually the charge of the whole Church as the Apostles had, but habitually onely by reason of the indefinitenesse of his office. He hath power in actu primo by virtue of his office, though not in actu secundo sive exercito, hee hath jus ad rem every where, but not in re any where, without a call. He is a Minister of Jesus Christ, and thereby hath right and power to perform the acts belonging to his office, but for the execution of it, there is required a call thereunto.
A couple more quotes from Hudson:
The division of the Church-Catholick into particular Congregations, seemeth to me to bee no further of divine institution, then as it fitly serveth for order and edification, by cohabitation, for injoyment of God's Ordinances together publickly (as the Jewish Church was divided by Synagogues, for their constant enjoyment of Word, praier, and discipline, which they could not constantly enjoy, as a Nationall Church, by their Nationall worship thrice in the year) and the same reason will by proportion carry it for Classicall, Provinciall, and Nationall divisions, for community of a greater part of the Church. (p. 17)
The nature of Synods is all one, whether they be Provinciall, Nationall, or Oecumenicall, and they only differ as greater or lesse, but their power in reference to their precincts, and delegation is alike. They differ from Presbyteries called Classes, because the Provinciall is constituted only of certain delegated members from the classical Presbyteries of the same Province; the National of delegated members from the Provincial Synods; and the Oecumenical of delegated members from the National Synods; whereas the Classis is constituted of the Elders of the particular Congregations combined together. The Classes are more frequent, constant, and ordinary in their meetings, the other more rare and extraordinary. The power of Synods is not at all civil, but Ecclesiastical, neither is it destructive to the power of Classes, or single Congregations, but perfective and conservative. They are not infallible, but may err as well as a Classis, or single Eldership, yet are not so subject thereto, because in the multitude of Counsellours there is safety, and they consist of more choice able men, and not so liable to personal prejudice against the accused, nor likely to be swaied by fear, or favour, or sinister respects. Their power is not meerly consultatory and suasive, but authoritative, and to be submitted unto by those for whom their delegation is, so farr as their acts are according to the Word of God. (pp. 158-159)
And here's a quote from Rutherford's Due Right of Presbyteries:
Synods are necessary for the well-being of the Church, and still are in the visible Church in more, or lesse degrees, for the authority of Synods consisting of six onely, differeth not in nature and essence, from a generall councell of the whole Catholicke visible Church. Magis et minus non variant speciem. And therefore if Synods be warranted by the word of God, (as no question they are) there is no neede to prove by particular places of the word, the lawfulnesse of every one of these, a sessionall meeting of the Eldership of a single Congregation. 2. A Presbytery, or meeting of Elders, or Pastors & Doctors of more Congregations. 3. A Provinciall Synod of the Presbyteries of a whole province. 4. The Nationall Assembly, or meeting of the Elders of the whole Nation. 5. The generall and Oecumenick Councell of Pastors, Doctors, and Elders of the whole Catholick Church visible, for all these differ not in essence, but degrees, and what word of God, as Matt. 18.16,17, proveth the lawfulnesse of one, is for the lawfulnesse of all the five sorts of Synods. (pp. 331-332 [second batch of page numbers])
These selections show, in contrast to modern popular opinion, that the historic Reformed faith does indeed hold to the concept of a formally visible, organic catholic church. Church authority does not reside only in the elders (the session) of individual congregations but is collegial in nature, so that church authority can (and should) be exercised not only in congregational sessions, but in presbyteries and higher synods (including provincial, national, and ecumenical synods). The formal catholic unity of the church and the collegiality of church power imply the impermissibility of both congregationalism and denominationalism (the idea that de jure ecclesiastical legitimacy and authority can reside in multiple, independent denominations or clumps of congregations).
For more, see here and here, and in general here.
UPDATE 6/4/14: In his earlier work on the nature and unity of the church, Essence and Unity of the Church Catholic Visible, found in the Anthology of Presbyterian and Reformed Literature, vol. 5, ed. by Chris Coldwell (Dallas, TX: Naphtali Press, 1992), Samuel Hudson quotes approvingly a description of the visible church from Lutheran theologian Johann Gerhard:
We have established the fact that the Church is a certain universal extended object, spread out through the orb, described to us in holy Scripture, which by means of a certain visible administration constitutes a singular ecclesiastical organic body, in which are comprised all the individual classical, provincial and national churches entirely as its parts.
UPDATE 6/9/14: Here is a well-articulated comment from church historian James Walker:
The visible church, in the idea of the Scottish theologians, is catholic. You have not an indefinite number of Parochial, or Congregational, or National churches, constituting, as it were, so many ecclesiastical individualities, but one great spiritual republic, of which these various organizations form a part. The visible church is not a genus, so to speak, with so many species under it. It is thus you may think of the State, but the visible church is a totum integrale, it is an empire. The churches of the various nationalities constitute the provinces of this empire; and though they are so far independent of each other, yet they are so one, that membership in one is membership in all, and separation from one is separation from all . . . This conception of the church, of which, in at least some aspects, we have practically so much lost sight, had a firm hold of the Scottish theologians of the seventeenth century.' (James Walker, The Theology and Theologians of Scotland. Edinburgh: Knox Press,  1982. Lecture iv. pp.95-6.)