No it wasn't. The GOP was born out of the Free-Soiler party, which subscribed to exactly that belief - that the territories should be admitted as free states. The Free Soiler's did indeed include SOME (but not all) abolitionists among their ranks, but not all free soilers were abolitionists themselves. When the GOP formed, the majority of its members were not abolitionists either.
which was essentially a church-based movement.
Some elements of abolitionism were church based. Others were not. Those that were included a very broad range of what could be considered churches - literally everything from fundamentalist christians to left wing civic-religion minded Unitarians. Most of the more prominent "religious" abolitionists were in this latter category, the unitarians, and had very little resemblence if any to what we would consider the "religious right" of today. Many of them were deists, pelagians, social-work Christians, and even a few agnostics.
At some point the abolitionists simply rejected the existing parties, mostly the Whig party, and formed their own.
That was true of the Liberty Party which existed in the 1840's, but it split in the early 1850's with many of its members going into the larger but not strictly-abolitionist Free Soil party. By the time the GOP came around, it included abolitionist and non-abolitionist elements alike as well as the old tariff whigs (of which Lincoln was one) and the various hamiltonian elements of American political thought.
It only won because the Southern Democrats basically boycotted the election
That too is not true. Every single southern state could have voted for the same candidate and Lincoln still would have won due to the northern population. His election was the product of the winner-take-all element of the electoral college more than anything else.
I find it hard to understand how anyone can make a "natural rights" argument in favor of slavery.
I don't believe anyone is trying to. There are several who are making a government-of-consent argument about the war though, and a solid majority of the southern population was denied that right by military coercion.
Had they successfully seceded, a later, much larger war would almost certainly have resulted as the two countries fought for control of the west.
How so? The southern states gave up their claims to the western territories save that of southern Arizona, which joined them in secession. The CSA states were even hesitant to allow Arizona to align with them because they knew that they had dissassociated themselves from all of the other territories.
Slave rebellions would have been viciously fought and ruthlessly put down.
The slave rebellions never happened as it was despite many, many attempts before and during the war to incite them.
No one at the time was under any illusion what free territories meant. The North already dominated the House because of its population. Free territories would eventually become free states with no corresponding slave state to keep the balance in the Senate, which would then also tip to Northern domination. Abolition would then become a possibility. Southerners and Northern abolitionists understood the battle over the territories was the battle over the future of slavery. That's why Bleeding Kansas bled.