During (and before) the Civil War, people like Abraham Lincoln had argued for this role, in fact. There was no right to secede, despite this right not being denied the states in the U.S. Constitution, because the Constitution itself presupposed certain conditions that were laid down by previous documents. The Preamble to the Constitution states that the purpose of the Constitution is to form a more perfect, i.e., a more complete, Union. More complete than what? More complete than the Union established by the Articles of Confederation. In other words: nothing in the Constitution can be interpreted in a way that would make the Union described in the Constitution less complete than the Union described in the Articles. The key issue here then becomes the Union as described in Article XIII:
Article XIII. Every State shall abide by the determination of the united States in congress assembled, on all questions which by this confederation are submitted to them. And the Articles of this confederation shall be inviolably observed by every State, and the union shall be perpetual; nor shall any alteration at any time hereafter be made in any of them; unless such alteration be agreed to in a congress of the united States, and be afterwards confirmed by the legislatures of every State.
The full name of the Articles of Confederation, in fact, is the Articles of Confederation and Perpetual Union.
The argument underlying perpetual union theory came to a judicial test after the Civil War, in the Supreme Court case, Texas v. White. Texas had received a large number of U.S. bonds in the Compromise of 1850, which among other things established the Texas border. These were sold off bit by bit in the years to follow, but by the time Texas seceded in 1861, there still were quite a few left. Although the U.S. decided not to honor any such bonds unless they were signed by Sam Houston (the pre-War governor), Texas managed to sell off quite a few of these bonds without any authorizing signature at all to George White, who ran a brokerage firm, and a number of these ended up being honored by the Union government after all. After the Civil War, Texas was put under Reconstruction, and this is where Texas v. White came in. Texas attempted to get the bonds back, claiming that they were sold illegally; White claimed that Texas could not establish this. At this point Texas was under three different governors -- a temporary transitional governor appointed by President Johnson (Hamilton), a military governor appointed by General Sheridan (Pease), and a governor who had recently been elected under Texas's new state constitution (Throckmorton). The Three Governors all authorized a lawsuit against White in order to regain the bonds. This case, of course, went to the Supreme Court, which had original jurisdiction to lawsuits involving a state as a party.
The primary argument of the state of Texas was that the Confederate legislature of Texas did not have the authority to authorize the sale of the bonds, because this was done for the purpose of supporting an illegal government (the Confederacy), and was therefore in violation of the U.S. Constitution. The defendants in response argued two major points. The first point was that the Supreme Court's original jurisdiction did not apply here. Texas, in seceding, ceased to be a U.S. state, and even at that point in time was not yet a state, but merely a territory in possession of the United States by military conquest (hence the Three Governors and the fact that Texas had not yet received representation in Congress again -- if I recall correctly, its state constitution still needed to be approved, or something like that). The second point was that the sale of the bonds was for the benefit of the people of the state and therefore was not illegal anyway.
The Chief Justice at that time, Samuel Chase (who had been a cabinet member under Lincoln), ruled as a matter of procedure that Texas at least had the right to prosecute the case. And his response appealed to the Articles of Confederation for part of the argument:
The Union of the States never was a purely artificial and arbitrary relation. It began among the Colonies, and grew out of common origin, mutual sympathies, kindred principles, similar interests, and geographical relations. It was confirmed and strengthened by the necessities of war, and received definite form and character and sanction from the Articles of Confederation. By these, the Union was solemnly declared to "be perpetual." And when these Articles were found to be inadequate to the exigencies of the country, the Constitution was ordained "to form a more perfect Union." It is difficult to convey the idea of indissoluble unity more clearly than by these words. What can be indissoluble if a perpetual Union, made more perfect, is not?
From this Chase derives the obvious conclusion:
When, therefore, Texas became one of the United States, she entered into an indissoluble relation. All the obligations of perpetual union, and all the guaranties of republican government in the Union, attached at once to the State. The act which consummated her admission into the Union was something more than a compact; it was the incorporation of a new member into the political body. And it was final. The union between Texas and the other States was as complete, as perpetual, and as indissoluble as the union between the original States. There was no place for reconsideration or revocation, except through revolution or through consent of the States.
No state, in other words, has a right to secede. By entering into the United States, Texas entered into a perpetual union. Thus any acts by which it attempted to secede were null and void. Therefore Texas, despite being in an illegal state of rebellion, never stopped being a state in the Union. It didn't have standard rights that the state had in the Union (like representation in Congress), to be sure, but it only didn't have them merely because it was in a state of violating the obligations upon which those rights depend, not because it had stopped being a state. As soon as regularity was restored, it would have all the rights of a state, because it had been a state all along, and the only thing preventing it from accessing its rights (so to speak) was that its relationship to the federal government was in an irregular state that needed to be fixed to make the exercise of those rights possible, and this only required Congress exercising its constitutional authority to suppress insurrection and guarantee the state a republican government. And this was being done, and the actions of the federal government to that point had already implicitly established that the government currently in power was a government authorized to represent Texas as a state.
Therefore Texas had the right to bring the case before the Supreme Court. Texas also won on the question of bonds; the sale was declared illegal, and Texas still had legal ownership of the bonds, and was entitled either to their return or to a cash equivalent.
Thus, by one of those ironies of history, the state which most confidently considers itself to have a right to secede -- and Texans broadly speaking do tend to believe in Texas's right to secede, despite having little desire to exercise it, and anyone raised in Texas has been taught it at some point -- was responsible for the court case that established, at least as far as interpretation of the Constitution goes, that no state has a right to secede.
Of course, things are a little more complicated than this. Chase's argument makes a number of assumptions that people have questioned -- whether the Articles really could guarantee perpetuity of union in the sense required, whether this perpetuity falls under the scope of the preambular "more perfect union," and so forth. It was a controversial decision at the time, and a narrow one (4-3). And being a matter of interpretive precedent rather than explicit statement in the U.S. Constitution itself, it has the limits all such precedent does. But it is there.