Can your gradual abiogenesis be falsified?
Yes, much in the same way as "macro" evolutionary theory can be falsified.
Can you make any predictions which, upon failing, would prove gradual abiogenesis never occurred?
We just had this discussion on this thread, and you finally conceded that we can conduct science on historical data.
"Life could be a natural process which could have been created by a supernatural intelligence."
True (and I am assuming you mean the formation of life could be by a supernaturally designed natural process), but that would still fit my definition of self assembly
Gradual abiogensis does not involve "self assembly" in any manner that I can understand. No creature or phyla, or any other agglomeration of breeding organic entities are "self assembled". That would require a time machine.
and contradict my assertion. If intelligence is THE process (whether intelligence is NATURAL or not is irrelevant)
The balderdash factor begins to loom up here. If the intelligent design was natural, than we are merely pushing the question back to--"Where did the natural designers come from?", which is pretty much the question we started with. If you weren't hiding a supernatural source of design in your back pocket, the issue wouldn't arise. A non-supernatural design explanation wouldn't actually impact modern evolutionary theory all the much--it would just inflate it from an earth-sized problem to a galazy-sized problem.
by which life can assemble from lifeless matter, then the natural process you propose does not exist (other than the natural process of intelligence). Otherwise it would falsify my claim. (Unless you want to argue that these hypothetical processes are intelligent in and of themselves.)
If natural life was demonstrably assembled by non-supernatural means, than those means were natural, and you have still done nothing to demonstrate that the particular way life was produced in your lab, was the only possible way, or even the likely way, life was originally produced. As I have pointed out, producing something you claim is "just like" life, is a far cry from demonstrating anything of statistical significance about actual life, much less provide any insight as to whether actual life's source is supernatural.