Quote:
. If two barrels of the same length where loaded with the same amount of powder and with projectiles of the same weight, but of different calibers, will then the stopping power of the guns be equivalent?

No.
The main reason being that "stopping power" is an unquantifiable thing. The "best" theories are not able to consistently predict and are only rough approximations of SOME observed behavior.
There is no math formula that gives any kind of reliable numbers for "stopping power" simply because EVERY case can have differing, often hugely differing factors.
People often use energy figures as indicators of "stopping power" but that is a flawed assumption as well.. Energy numbers are a math calculation, and DO provide a consistent medium for comparison between dissimilar calibers but are not valid regarding "stopping power" because nothing really is.
Consider this, just as an example.
You can load a .22250 and a .4570 to identical energy levels (ft/lbs).
Energy (alone) says they are equal. Both will certainly kill, but which one would you choose to stop an 800lb angry beast who wanted very seriously to stomp you to bits??
Every animal is different, every person is different, every SHOT is different. Often the differences are small and unnoticed. Sometimes they are large and the determining factors.
There is an old African saying that goes something like this...
"one day, you meet a lion on the trail, and he runs away...
The next day, you meet his brother, and your family wonders why you did not come home for dinner...."
Your example of equal weight bullets with equal powder pushing them only difference being diameter will give equal results using math, but CAN show vastly different results in the field due to all the other factors involved.
My signature line is what I have personally seen. I hold it to be essentially true.