I did read what you wrote - you didn't define how he had been proved to have killed someone, via legal process or otherwise - if by legal process I don't see the relevancy of the baying mob.

If proven by legal process then an execution would be perfectly democratic - not because 'a thousand people are howling "hang him"' but because the law, decided upon in a democratic way, stated that was the punishment to be applied.

Those protesting that decision would also be behaving in a democratic way by exercising their right to free speech and attempting to persuade others to their point of view.