The fundamental cause of southern secession (and ultimately the Civil War) was the US's inability to solve slavery at the national level.

The Civil War was not fundamentally about "states rights". Asserting a state's right to secede doesn't speak to why the state wants to secede. Steven's citation of reasons in his answer only serve to underline this. When the northern states were threatened by the War of 1812, they considered secession. When South Carolina was threatened by a tariff, they attempted to nullify the law. When a state's self interests come into play, they'll take advantage of whatever political mechanism they can imagine to assert that self interest, up to and including nullification, secession, and war.

**The real story is not in the political mechanics but the underlying interest in preserving slavery that forced the South to become so hell bent on their "states rights".** If Northern states had seceded over the War of 1812, we wouldn't assert the fundamental cause was a debate over state's rights. Rather we'd say it was their opposition to the War of 1812. The same applies for the South's secession as well. Their interest in preserving slavery drove them to use untested constitutional mechanism and eventually go to war.

And when you consider the South's conundrum it becomes clearer.

White southerners lived in constant, real fear of slavery/black insurrection. They had experienced violence from slaves during [Turner's Rebellion][1]. They looked south to the [Haitian Revolution][2] and similar rebellions in the sugar colonies and saw little comfort in how those societies were transformed. Maintaining the institutions associated with slavery was a matter of life/death for white southerners. [Laws were passed][3] to further and further restrict the activities of slaves and freed blacks and codify racial distinctions. Even if the injustice of the system might be acknowledged by some southerners, the fear that slaves or freed blacks might (perhaps justifiably) seek vengeance was deep.

Even if a southerner thought that ending slavery was the right thing to do (and many did), the way to get there without providing massive disruption to southern society (economic or otherwise) was very hard to see. Aside from what they would have perceived as lost property, you have the question of [what do you do with the slaves][4]? If they're freed won't they just take up arms and overthrow southern society? Would the freed blacks compete with poor whites and drag down wages? In every sense, security, agrarian economics, social interactions, slavery and the associated racial class system was at the heart of Southern life and it was apocalyptic to imagine life without it. Lincoln himself [points out this fear][5] during his inaugural speech when trying to be conciliatory

>Apprehension seems to exist among the people of the Southern States that by the accession of a Republican Administration their property and their peace and personal security are to be endangered.

Indeed clauses to maintain of the institution of slavery are rife throughout the [Confederate constitution][6]:

Article I, Sec 9:

> No bill of attainder, ex post facto law, or law denying or impairing the right of property in negro slaves shall be passed.

Article IV, Sec 2 (1):
>  The citizens of each State shall be entitled to all the privileges and immunities of citizens in the several States; and shall have the right of transit and sojourn in any State of this Confederacy, with their slaves and other property; and the right of property in said slaves shall not be thereby impaired.

Article IV, Sec  2 (3):
> No slave or other person held to service or labor in any State or Territory of the Confederate States, under the laws thereof, escaping or lawfully carried into another, shall, in consequence of any law or regulation therein, be discharged from such service or labor; but shall be delivered up on claim of the party to whom such slave belongs,. or to whom such service or labor may be due.

Article IV, Sec 3 (3)
> The Confederate States may acquire new territory...  In all such territory the institution of negro slavery, as it now exists in the Confederate States, shall be recognized and protected be Congress and by the Territorial government;

This need for security and tight adherence to the institution existed in direct contrast to Northern abolitionist's deep, often religiously founded, moral opposition to slavery from people such as [John Brown][7] and [others][8]. Many movements sprung up to curtail the expansion of slavery such as the [Free Soilers][9] and [Republicans][10].

The national politics prior to the civil war became dominated by the South's attempt to keep slavery going and expanding vs the North's determination to keep it from expanding. Southerners understood that the only way to maintain their power in Congress (and ultimately their sense of security) and to grow economically was to fight for the expansion of slavery. The South would uses the power of the Federal government to protect and maintain slavery. The [Fugitive Slave Act][11] along with the [Dred Scott decision][12] show that the South is happy to use/laud Federal institutions if it helped them meet their end goal. In fact many feared that these were overreaches of Southern influence into Federal power. To quote Lincoln's [House Divided speech][13]:

>We shall lie down pleasantly dreaming that the people of Missouri are on the verge of making their State free, and we shall awake to the reality instead, that the Supreme Court has made Illinois a slave State.

Many in the North resented the power that the 3/5ths compromise, the Dred Scott decision, the Fugitive Slave Act, the annexation of Texas, and the Mexican-American War gave to Southern states and argued that the South had an inordinate amount of power. Indeed Lincoln above is close to asserting a "states right" to deny slavery. In addition some Abolitionists themselves [considered secession][14] due to their perception that the South had a large amount of power.

Both the sectional conflicts around expanding slavery and the issue of slavery were ultimately irreconcilable. While campaigning Lincoln [attacked][15] the South's hold on Federal power through the continued expansion of slavery through warfare, annexation, law and the 3/5ths compromise. After Lincoln was elected, there was already a shooting war in [Kansas][16] and [John Brown][17] had raided Harpers Ferry in an attempt to incite a massive slave revolt. Lincoln's party was seen by the South as radically anti slavery, and indeed [opposition to slavery was the issue that their party was founded on][18], to quote this [history of the Republican party][19]:

> The Nebraska Bill passed the house on May 22. The next day about thirty anti-slavery members of the House of Representatives, Whigs and Democrats, held a meeting and discussed the necessity of organizing a new party under the name "Republican,"and they pledged themselves to fight against the extension of slavery.

The South's siege mentality would only reach a fever pitch upon Republican Lincoln's election. With what they perceived as a virulently anti slavery, pro Northern party increasing its influence coupled with and ever increasingly violent political far-left political opposition, the south felt deeply threatened politically, economically, and physically. Virginia's [secession ordinance][20] while brief, discusses the "oppression of the Southern slaveholding states" 

> ... the Federal Government having perverted said (costitutional) powers, not only to the injury of the people of Virginia, ubt to the oppression of the Southern slaveholding states.

Also as [this blog][21] summarizes Civil War historian James McPherson:

> White southerners, he argues, saw the Lincoln administration and the Republican Party as the revolutionaries. The move to secede was a counter-revolution, a conservative  effort designed to protect what they had and stem the tide of change sweeping across the nation. All of their resistance, he argues, was aimed at maintaining slavery and their position in society

They tried whatever mechanism they had, citing state's rights and eventually finding themselves in a war. Without doing so, they feared, they would have change forced on them through incited slave rebellion or what they perceived as revolutionary Republicans. Even if the end of slavery wasn't imminent, the loss of the political power that had been protecting the institution felt deeply threatening due to the magnitude of how fundamental slavery was to southern life. So threatening that they were willing to use whatever constitutional mechanism necessary to preserve their political independence and keep slavery going.

**Edit in response to Steven's Edits**

The issue isn't whether legally the South had anything to fear at the legal level. The issue was that they had managed to use Federal power during the antebellum period to allow the continued spread and protection of slavery. Despite deep fissures, they had managed to continue to dominate and grow political power to protect their slave holding society. The incoming Republicans threatened that as [wikipedia][22] puts it

> began as coalition of anti-slavery "Conscience Whigs" and Free Soil Democrats opposed to the Kansas-Nebraska Act, submitted to Congress by Stephen Douglas in January 1854. 

[Also their response to the Nebraska Act][23]

> The Nebraska Bill passed the house on May 22. The next day about thirty anti-slavery members of the House of Representatives, Whigs and Democrats, held a meeting and discussed the necessity of organizing a new party under the name "Republican,"and they pledged themselves to fight against the extension of slavery.

Its understandable Lincoln would be fairly conciliatory in his inaugural speech. The south was taking action to secede. He wanted to maintain the Union and preferred a peaceful solution. He also has to worry about the slave holding states that remained in the Union. That doesn't change the platform of his political party, the reason the South found him so threatening politically. Lincoln himself [acknowledges][24] how apprehensive the South is to the Republican takeover:

> Apprehension seems to exist among the people of the Southern States that by the accession of a Republican Administration their property and their peace and personal security are to be endangered.

He echoes my assertion of how much upheaval white southerners saw in ending slavery.

Regardless of any political or legal institutions, when 1/3 or more of your population is enslaved, you might worry more about the plans of those enslaved. When someone like John Brown comes knocking at your door to incite a revolution, you do what you have to to baton down the hatches, especially when it seems his incendiary, potentially threatening actions appear to be [hailed in many in the North][25]. No law is going to prevent the enslaved from taking back their freedom when given the opportunity.