Darwin did propose that all extant organisms have a common ancestor:
Therefore, on the principle of natural selection with divergence of character, it does not seem incredible that, from some such low and intermediate form, both animals and plants may have been developed; and, if we admit this, we must likewise admit that all the organic beings which have ever lived on this earth may be descended from some one primordial form. But this inference is chiefly grounded on analogy, and it is immaterial whether or not it be accepted. No doubt it is possible, as Mr. G.H. Lewes has urged, that at the first commencement of life many different forms were evolved; but if so, we may conclude that only a very few have left modified descendants. For, as I have recently remarked in regard to the members of each great kingdom, such as the Vertebrata, Articulata, etc., we have distinct evidence in their embryological, homologous, and rudimentary structures, that within each kingdom all the members are descended from a single progenitor.
--Darwin, Origin of Species
He probably wasn't the first one; Darwin cites a previous suggestion:
From a circular lately issued it appears that Dr. Freke, in 1851 ("Dublin Medical Press", page 322), propounded the doctrine that all organic beings have descended from one primordial form. His grounds of belief and treatment of the subject are wholly different from mine; but as Dr. Freke has now (1861) published his Essay on the "Origin of Species by means of Organic Affinity", the difficult attempt to give any idea of his views would be superfluous on my part.
--Darwin, Origin of Species
As you can see, he offers no real evidence for this; he explicitly says it's speculation. Of course, nothing from Darwin is necessary for modern evolutionary theory, and other groups have tried to make this more concrete. Several lines of evidence provided increasing evidence for a universal common ancestor (UCA):
For a century after the publication of Darwin's bold proposition, before the advent of molecular biology, the UCA hypothesis remained an untested and hardly testable speculation. However, first the universality of the genetic code and later the demonstration of the (near) universal conservation of approximately 100 RNA and protein-coding genes among cellular life forms provided ample evidence in support of the UCA. Although generally considered compelling, this evidence fell short of a rigorous, formal test of the UCA hypothesis.
--The common ancestry of life
A relatively recent approach has tried using information theory to formally test a UCA:
Here I provide the first, to my knowledge, formal, fundamental test of UCA, without assuming that sequence similarity implies genetic kinship. I test UCA by applying model selection theory to molecular phylogenies, focusing on a set of ubiquitously conserved proteins that are proposed to be orthologous. Among a wide range of biological models involving the independent ancestry of major taxonomic groups, the model selection tests are found to overwhelmingly support UCA irrespective of the presence of horizontal gene transfer and symbiotic fusion events. These results provide powerful statistical evidence corroborating the monophyly of all known life.
--A formal test of the theory of universal common ancestry
Formal positive proof of theories in science, outside of math, are rare to non-existent (what actually happens is that competing theories are disproved) and I don't think that this approach is different. It's not a formal proof, it's yet more powerful evidence that theories not involving a UCA suck.
A formal demonstration of the Universal Common Ancestry hypothesis has not been achieved and is unlikely to be feasible in principle. Nevertheless, the evidence in support of this hypothesis provided by comparative genomics is overwhelming.
--The common ancestry of life
