Chomskian view on grammar The Chomskian view asserts that certain structural components of grammar are innate to all humans. To support the Chomskian view, the observation that we are able to omit certain pronouns in the English language without ever being taught the grammatical rules necessary is used to support the view. However, my question is, what would an example of the above observation be?
Summary:
It's worth noting that all languages have an oral origin so concerning the biological origins of grammar we must look for neurological structures involved in speech processing/production. However, it's worth first clarifying Chomsky's view that there are two levels of innateness: at the level of genetics i.e. biological heredity and at the level of neurological structures i.e. morphology and morphogenesis.
Based on the available evidence, it appears that specific genes and neurological structures that aid language acquisition do exist but these aren't necessarily specific for language.
 Chomsky's view:
Chomsky has historically held firm to the view expressed in 1983 [1]:

QUESTION: Is the role of heredity as important for language as it is
for embryology?
CHOMSKY: I think so. You have to laugh at claims that heredity plays
no significant role in language learning because exactly the same kind
of genetic arguments hold for language learning as hold for
embryological development. I’m very much interested in embryology but
I’ve got just a layman’s knowledge of it. I think that recent work,
primarily in molecular biology, however, is seeking to discover the
ways that genes regulate embryological development. The gene-control
problem is conceptually similar to the problem of accounting for
language growth. In fact, language development really ought to be
called language growth because the language organ grows like any other
body organ.

However, Chomsky also recognised that there was little neurobiological evidence to support this view at the time beyond reasonable hunches:

QUESTION: Is there a special place in the brain and a particular kind
of neurological structure that comprises the language organ?
CHOMSKY: Little enough is known about cognitive systems and their
neurological basis; so caution is necessary in making any direct
claims. But it does seem that the representation and use of language
involve specific neural structures, though their nature is not well
understood.

Chomsky's particular hypotheses:
From the above interview we may derive two hypotheses:

That there might be genes for linguistic ability.
That the development of language may rely on neurological structures that are specific for language.

The first hypothesis is partially validated by research on the FOXP2 gene [3]. In 2001 it was found that in humans the mutation of this gene significantly disrupts speech and language skills.
That said, we must note that FOXP2 has other roles:

...FOXP2 is also expressed in defined regions of other tissues during
embryo development, including the lung, the gut and the heart, as
well as in several tissues of the adult organism.

Given that the heart, gut and lung developed prior to linguistic ability it's more likely that FOXP2 was initially useful for heart and lung function, and was exploited for linguistic ability rather than the other way around. This gene is actually present in other animals. The authors of [3] clarify this point:

the presence of FOXP2 in other animals does not diminish its relevance
for speech and language, but rather represents another example of
recruitment and modification of existing pathways in evolution.

The possibility of specific neurological structures:
Regarding the second hypothesis, this is the subject of ongoing investigation that combines AI research, cognitive science, neuroscience and morphogenesis. How language develops is a very complex and fascinating subject.
In a pioneering robot experiment [2], the linguist and cognitive scientist, Pierre-Yves Oudeyer showed that with basic neural equipment for adaptive holistic vocal imitation, coupling directly motor and perceptual representations in the brain, it was possible to generate spontaneously shared combinatorial systems of
vocalizations in a society of babbling individuals:

...the transition from inarticulated vocalization systems to
human-like speech codes may have been largely due to a modest
biological innovation. Indeed, the model indicates that neuronal
structures that encode a priori and specifically phonemic
organization, as well as typical regularities of speech, do not need
to be innately generated to allow the formation of such speech code.
This is the second contribution of this work: it allows us to
understand how the self-organizing properties of simple neural
structures may have constrained the space of biological vocalization
structures and how speech codes may have been generated and selected
during phylogenesis. These new hypothesis may not have been identified
without the use of computer models and simulations, because the
underlying dynamics are complex and difficult to anticipate through
uniquely verbal reasoning. This illustrates the potential importance
of these new methodological tools in human and biological sciences.

Pierre-Yves' research on robot languages suggests that:

The specific neurological structures that aid acquisition aren't necessarily specific for language.
Cognitive scientists should investigate the development of the vocal tract during language acquisition.

Concerning specific neurological structures, today we know that Broca's area is heavily used for speech production but similar regions are present in other apes that have limited linguistic ability [6]. So it's important to consider the causal influence of Broca's region within cortical networks involved in speech production. In [7] the authors share that:

Using direct cortical surface recordings in neurosurgical patients, we
studied the evolution of activity in cortical neuronal populations, as
well as the Granger causal interactions between them. We found that,
during the cued production of words, a temporal cascade of neural
activity proceeds from sensory representations of words in temporal
cortex to their corresponding articulatory gestures in motor cortex.
Broca’s area mediates this cascade through reciprocal interactions
with temporal and frontal motor regions.

For this reason in the discussion section they infer:

...Broca’s area is not the seat of articulation per se, but rather is
a key node in manipulating and forwarding neural information across
large-scale cortical networks responsible for key components of speech
production.

So far there is a lot of neurological evidence supporting Oudeyer's conjecture that specific neurological structures that aid acquisition aren't necessarily specific for language. As for Oudeyer's second suggestion there's still a lot we don't know. You might be interested in the work of the Max Planck Department for the Neurobiology of Language[4,5].
 References:

Noam Chomsky. Things No Amount of Learning Can Teach. Interviewed by John Gliedman. 1983.
Pierre-Yves Oudeyer. Self-Organization and Complex Dynamical Systems
in the Evolution of Speech. 2013.
Gary F. Marcus and Simon E. Fisher. FOXP2 in focus: what can genes tell us about speech and language? Elsevier. 2003.
Hagoort, P., &amp; Beckmann, C. F. (in press). Key issues and future directions: the neural architecture for language. In P. Hagoort (Ed.), Human Language: From Genes and Brains to Behavior. Cambridge, MA: MIT Press.
Hagoort, P. (in press). Introduction. In P. Hagoort (Ed.), Human Language: From Genes and Brains to Behavior. Cambridge, MA: MIT Press.
Claudio Cantalupo and William D. Hopkins. Asymmetric Broca’s area in great apes. NIH Public Access. 2001.
Adeen Flinker, Anna Korzeniewska, Avgusta Y. Shestyuk, Piotr J. Franaszczuk, Nina F. Dronkers, Robert T. Knight, and Nathan E. Crone. Redefining the role of Broca’s area in speech. PNAS. 2015.