Despite their rhetoric about the permanence and indestructibility of the Union, both Lincoln in his First Inaugural, and the Supreme Court in Texas v. White, strongly implied that it would be possible for one or more states to leave the Union with the consent of the Union as a whole.
By what legal mechanism would such secession through mutual agreement be accomplished? The most obvious answer is a statute enacted by Congress. Just as Congress can approve the admission of new states, the argument would go, so it can let old states leave.
Texas is the only state that can secede.
Among the Founding Fathers there was no doubt. The United States had just seceded from the British Empire, exercising the right of the people to “alter or abolish” — by force, if necessary — a despotic government.
As did James Madison. I think that the implication is that a state may leave by the same manner as states join - a simple majority vote in both Houses of Congress. No super majority, no amendment, just a vote.
I don't think it would be a legal issue, but rather, a military issue.
The BIG answer, provided by U.S. Supreme Court decision in 1869:
Unilaterally, they cannot, just as states cannot unilaterally enter the union on their own accord. Both acts require the consent of the Union as a whole. Specifically, the final step of whether a state is admitted or withdrawn from the union (or whether it is merged into another state or split into multiple states, or separated from another state) is determined by act of Congress.
The Constitution clearly states that members of the federal government, as well as state legislatures, are bound by oath to support the U.S. Constitution, and that they cannot form any confederation with another state without consent of Congress first. States cannot wage war or take up arms against the United States (this clause was provided because the original government, the "articles of the confederation" that the founders created from 1774-1789, allowed states to ignore the federal government ruilings and take up arms against the government, and it proved to be a disaster)
States agreed to these terms (that they were bound by oath to stay loyal to the U.S. Constitution and could not unilaterally create their own rival government) before they agreed to enter the union. States who violate these terms, (like South Carolina attacking U.S. military installations and seizing federal property without consent of Congress) are engaged in armed insurrection against the United States and the rebellion can be put down accordingly.
The United States did not "secede" from the United Kingdom, since the United State was not a state within the U.K. Their are 86 Counties (their equivalent of "state") in the Great Britain, and none of the 13 colonies in North America was one of them. The last time any the 86 counties were engaged in insurrection against the government was the British civil war in the 1600s (the American colonies did NOT participate in that skirmish, but since the anti-royality faction won, we were briefly governed by a new nation, the short lived "Commonwealth of England" from 1649-1660)