Darwin's theory is this:
Observation 1: Most individuals produce far more offspring than is needed to replace themselves.
Conclusion 1: Most individuals must die without reproducing.
Observation 2: In any population, there is a wide range of variation.
Assumption 1: Those individuals that survive and reproduce will be the ones whose characteristics make it more likely that the individual will survive.
Observation 3: Much of this variation is hereditary.
Conclusion 2: This "natural selection" will cause certain characteristics to decrease in frequency, perhaps even disappear completely; and cause other characteristics to increase in frequency, perhaps to become evident in every member of the population.
Observation 4: Humans have bred many varieties of plants and animals with characteristics that are not observed in the wild ancestral species.
Conclusion 3: There is a source of new heretable characteristics.
Conclusion 4: The production of new characteristics with natural selection will cause a species to undergo a tremendous amount of change over very long time periods.
Observation 5: For each domesticated species, humans have bred a large number of different breeds.
Observation 6: Life can be classified in a heirarchical system.
Conclusion 5: All current (and known past) species evolved from a small number of, perhaps only a single, ancestral species.
Note that there is no assumption whatsoever about the structure of the cell, or even that organisms are composed of cells.