Can the retina emit light?
Please allow me to start by saying I know nothing about LEDs, so my answer isn't going to address your analogy. I'm also going to keep it more simple.


  Can the retina emit light?


Yes. The retina is able to give off light in the form of fluorescence. Since fluorescence is the emission of light by a substance (granted that substance must first absorbed light or other electromagnetic radiation of a different wavelength.)


  “We can use autofluorescence emission to image the retina and see certain features of it,” said Dr. Sparrow, the Anthony Donn Professor of Ophthalmic Science in the departments of Ophthalmology, and Pathology &amp; Cell Biology. ...Most objects in the environment are visible because they reflect, as opposed to emit, light. However, the retina—which exists within the eye’s back wall, known as the fundus—can generate its own fluorescence.


The cells of the deepest layer of the retina, called the retinal pigment epithelium (RPE), contain fluorescent compounds, which when excited by blue light emit a yellow fluorescence.

This ability changes with age, disease, etc. 


  Excessive accumulation of lipofuscin granules in the lysosomal compartment of retinal pigment epithelium cells represents a common downstream pathogenetic pathway in various hereditary and complex retinal diseases, including age-related macular degeneration.


Lipofuscin is one of several fluorophores in the RPE. Different fluorophores may be related to different damage to the eye.


  A major hydrophobic component of RPE lipofuscin is the fluorophore A2E... Given that A2E strongly absorbs in the blue region of the spectrum, together with the known susceptibility of RPE to blue light damage, we undertook to investigate A2E as a fluorophore involved in blue light toxicity. 


Science Insight: Using the Retina’s Natural Fluorescent Light to Measure and Treat Disease
FUNDUS AUTOFLUORESCENCE IMAGING: Review and Perspectives
The Lipofuscin Fluorophore A2E Mediates Blue Light–Induced Damage to Retinal Pigmented Epithelial Cells