He's comparing apples and oranges here.
Given that avg. radius of the earth is 6,400km, and that of the atmosphere is by the Karman line (110km), you have the equivalent of 110km/6400km or 1.7cm%. So far, so good.
The trouble is when you force fit your other data on the apple side.
Apples range from a size 64 to a size 216 (http://stemilttrade.com/wp-content/uploads/2013/07/AppleSizing.pdf). All of these would be cited as a "ordinary" apple.
Skin thickness also sorts out from a low of 33um up to 76um, with variation between different thicknesses on the same apple. (http://www.agriculturejournals.cz/publicFiles/51351.pdf)
Putting these two apple facts together, we get a broad range of values - from4.3cm4.6 cm radius down to2.31cm2.9 cm radius, combined with.033cm0.0033 cm to.076cm0.0076 cm --> yielding a low of.00760.00072 or 0.072% and a high of.02390.0026 or 0.26% ... way outside our range.
Could you find some set of apples where his example would be true? Yes. Good luck with that.
The take home lesson is not to make fruit analogies for planets.
Sorry, but you also seemed to have some errors with your units and conversions.
First correction, I assume you meant to express what should have been a dimensionless ratio as a percentage.
I'm not sure what happened when you converted apple diameters in inches to radii in centimeters. I think that is what you were attempting.
Also, a micrometer is 1/10,000 of a centimeter, I believe, and not 1/1000.
If you got your apple skin sizes from the abstract of the pdf file to which you linked, then your 76 um probably should have been a 73 um, too, but I let that slide.
I wouldn't want to add to the confusion. ...lol