I don’t know that it would be worthwhile to make a law even if it ought to be (which I doubt).
If the hypothetical billionaire were truly concerned with the global population, he would focus his contributions more intelligently and effectively per birth avoided. There are many locations world-wide that are underserved when it comes to birth control. Or, if his worries were targeted at per capita GHG emissions rather than simply births, he could offer targeted financial incentives in either the highest per capita emitting nations or those emerging economies having the sharpest population growth curve.
Either way, $10,000 per birth avoided is inefficient. It’s overpaying by one to two orders of magnitude.