Please allow me to point out some flaws in your argument.
First, comparison of the world-wide birth rates vs. death rates is not useful because it does not take into account the rates relative to the level of societal education and development. In "Western" countries the population growth rate is close to zero or even negative. So your argument only has a basis (population-growth) in countries which do not have access to birth control or countries where women do not have rights or means to prevent conception.
Second, your argument fails to take into account technology development. The question could be rephrased as "What is the carrying capacity of the earth?" which could further be rephrased as as "What is the carrying capacity of a robust molecular nanotechnology enhanced earth?". Your argument only has merit if all people continue to reproduce at world-wide rates and if nanotechnology fails to develop. Assertions that I would claim you cannot make.
Now if you become informed about the real limits to a greater extent and wish to discuss them, then I would be happy to engage in such a discussion.
1. Q1 to bring to the discussion: "What is the hypsithermal limit of the Earth and why is it significant?" 
2. Bonus points: How does one escape the hypsithermal limits of the Earth?"