What you are saying basically is let humans die naturally. Do not use medication right? Sounds alright until you yourself get a decease. Science has come a long way since the age of the religions which taught what you are saying. You are highly likely be dead by now if it were up to the nature entirely.
Yes we have upset the global eco system but now we are working on reversing the damage and minimizing the future damage. We are not using CFC anymore. We do not go out and kill wild animals in a mass scale for fun like a few decades ago. In a global scale, we are slowly making progress. So no need to let people die of cancer and other horrible deceases don't you agree?
Here are a few facts as well

1. Global annual population growth rate is decreasing rapidly. Even though right now the population is growing, that rate will reduce eventually (Check this out https://ourworldindata.org/world-population-growth/) Yes it will take time but we are finding more and more sustainable ways of living every day thankful to science.
2. Actually the effect of natural killers on the human population growth is negligible when we consider all the reasons causing death to humans. You are far far more likely to get hit by a vehicle than killed by an animal. As for now the major causes of death are noncommunicable deceases. You can see some numbers on this if you take a look at WHO's data (Here's a one I found http://www.who.int/mediacentre/factsheets/fs310/en/)