So I had a blowout on my front tire after hitting a fairly large pothole and immediately pulled into a side street and proceeded to change the front tire and put on the spare. The manual specifically states not to mount the spare on the front as the diameter of the wheel is smaller than the stock tire. I was in a hurry and drove on it for ~3 miles, then switched the rear tire with the spare at home. Could this have caused any damage to my differential? BTW, the blowout was from a previously patched hole. Vehicle is a 17' Honda Civic.
It is extremely unlikely that you caused any damage whatsoever in the three miles you drove.
Basically, there are a couple little gears that don't normally spin except when you go around corners. When the tires are different sizes they spin all the time.
Three miles isn't enough to cause these gears abnormal wear, especially if you weren't driving it hard.
For a detailed explanation of how a differential works, see the Wikipedia article
The gears which spin when you run different sized tires are called spider gears, and are tagged number 4 in this diagram.
The Traction Control System (TCS) error you saw displayed was caused by the tires spinning at different rates. Traction control detects differences in the rate of rotation of the wheels, and assumes that if they're spinning at different rates one or more of them has lost traction. In your case, one of the wheels was smaller, so it was always spinning at a different rate.