I do think he has a point though. Wherever people are wealthy enough to spend money on vacations and other activities, they seem to value leisure time over reproduction.

Currently, the world population is still growing, but that's mostly due to poorer countries and sub-populations. AFAIK, not even the US's whites are enough to sustain the population size:

"Expected lifetime births have also fallen for White non-Hispanic women from 1.91 in 2007 to 1.75 in 2013. U.S. White non-Hispanic births have been below the replacement level of 2.1 births per woman for decades." (http://www.unomaha.edu/news/2015/01/fertility.php).

It is even worse in Europe and Russia.