You are assuming that everyone is entitled to an equal share of the wealth of the world, irrespective of who produced it, and why.
First, this is at least as strong an assumption as the just world hypothesis, and at least as poorly supported. As it happens, invoking the "just world fallacy" is usually a sign that one is about to commit a fallacy that is just as bad. In fact, I would say it is worse, from a series of theoretical and practical considerations.
In my previous comment I have already shown an example of how your hypothesis (I am going to be charitable and not refer to it as a fallacy) leads you to a paradox. But there are many more.
Consider people born at different times of history. Do people who were born in the past suffer an injustice from the people who live in the present? "Well, but I can't do anything about it." Not for the ones who were born two hundred years ago, but what about fifty? You are clearly a young guy. Maybe even more of your income should be taxed and given to the baby boomers, to compensate them for the injustice of having been born in an age without the internet. (This is starting to look like a just world hypothesis applied to the system of defined benefit pensions...)
It gets worse. Demographically, socialism is the ideology of bourgeoise youths who hate their parents. Psychologically, its root is the rejection of one's parents, and of the advantages received from them. Therefore, guilt for "unearned privilege". But while it is true that the child did not earn the privilege, it is not true that it was not earned. It was not earned by the child, but it was earned for him, by his parents. (It is not a coincidence that adults, and especially married people, are more likely to be conservative.)
So try putting yourself in the shoes of the people whose work produced that wealth, which their children now inherit. Do they not get a say in who gets it? (Remember that they are already suffering from the injustice of being born in the past, so by denying them a say we are adding injustice to injustice.)
And of course, it does not stop at material wealth. There are all sorts of advantages (or disadvantages) one inherits from one's parents, in terms of education, social network, genetics, and so on. The family is the fundamental unit for the generation and transmission of inequality. Which is why the early socialists, who were intellectually honest, made it their explicit program to abolish the family.
Of course, there are many social problems with destroying the family. But there are also economic problems! Being able to transmit wealth to one's progeny is an extremely strong incentive for economic activity and growth beyond pure self-sustenance. In fact, property and inheritance are universal to all developed human civilizations: all of those proto-communist tribes anthropologists found never got much further than the stone age. The most likely conclusion is that they cannot: property and inheritance (in other words, inequality) appear to be an essential social technology that is required for development. The best a communist can hope for is that they may become unnecessary at some later stage of technological super-abundance, when we can all go off and play cowboys in space.
But we ended up focusing on economic factors again. Your idea of equality has much broader implications, though. If you say that the luck of one's birth is injustice, you cannot limit that to wealth. Differences in intelligence are unjust. Differences in attractiveness are unjust. We could go on and develop all the implications, but I'm going to stop here.