"The choice fell on me" sounds odd, as the more common idiom is "The choice fell to me", meaning that I was the one required to choose.
Since no one else would do it, the choice fell to me to decide what color to paint the house -- a thankless task, since no matter which color I picked I knew someone would object.
However, I don't think it's wrong as much as rare. I found a couple of results in older scientific journals:
In regard to Streptomycin the committee's choice fell on Selman Waksman, the bacteriologist who had ascertained the effect unambiguously and also published it.
For insulation, the choice fell on expanded rigid polyvinyl chloride, usable in sandwich form ...
From context it's clear the phrase means that whatever the choice fell on is what was chosen, not what did the choosing.
However, I recommend against using this idiom as it may be confusing. When no actual choice is involved, you can use the more common, "it fell on me" (as in your linked definition) to do something, meaning that you were forced to do it by lack of other options.
After a week I couldn't even see the kitchen counter for all the piled-up dirty dishes, and we were completely out of clean ones. Since clearly none of my roommates would do it, it fell on me to do the washing-up if I didn't want to live in squalor. But I vowed I'd make them pay for it ... somehow, someday.