How can I wire a recessed light from an existing remote controlled fan?
You didn't say what jurisdiction you are in - electrical codes do differ by location.

You will also need a neutral and a ground for your fixture.  You should fish a standard 14/2 wire for that.  In my area, all neutrals and all grounds should be combined in the box with wire nuts, so wire the neutral (white) and ground (bare) from the 14/2 wire that you fished to the nearby light, and connect the black wire from the 14/2 to the red wire you are referring to.

I am assuming that you would not be overloading your circuit by adding a new device (fixture).

Also, I would suggest you have an electrical inspection.  If your house burns down due to electrical fault in uninspected work you did, then your insurance could possibly deny any claim. :(

(In 2011, I rewired my 1959 house and brought it up to then-current code. And yes, I had it inspected.)