This is obviously untrue at a a global scale assuming the usual measure of crude death rates per year, which around 8 deaths per 1,000 people per year. The world population keeps rising for now despite that.
Gates also doesn't specify a bound for the lag in his implication, e.g. 100 years or just a few years lag between cause and effect? Claims this vague they can be proven true by stretching them enough. The rate of growth of the world population is decreasing, so given enough (lag) time Gate's "prediction" might prove "true".
Generally speaking, lower mortality is associated with higher GDP per capita (see first link), which in turn is associated with lower natality. Some researchers even dispute the latter, finding a J-shaped curve if the Human Development Index is substituted for GDP/capita. (There's a temporal counterpart to that. The number of countries with a fertility rate below 1.3 had peaked in 2003 [over 20 countries], but that number was only around 5 in 2008.)
Without any numerical landmarks, what Gates is talking about is called the Demographic Transition Theory, and goes back 50-70 years:
The mainstream arguments of the theory are that fertility is high in
poor, traditional societies because of high mortality, the lack of opportunities for individual advancement, and the economic value of children.
All these things change with modernization or urban industrialism, and
individuals, once their viewpoints become reoriented to the changes that
have taken place, can make use of the new opportunities.
Wikipedia also has a rather verbose page on it.
In general researchers who have tried to quantify this theory and use it for predictions have been proven pretty wrong.
Though he was by no means the first to state the essentials of the theory of the demographic transition, Notestein early formulation is conventionally accepted as classic. [...] Notestein thought that the populations of Western and Central Europe would peak in about 1950 and decline thereafter. The corresponding date for Southern Europe was 1970. Like Thompson, Notestein assumed that fertility would fall more steeply than it did in fact. His estimate of the total world population in the year 2000 was 3.3 billion
in contrast to today's  expected figure of nearly six billion.
Even with the hindsight of 50 years it has not been resolved whether the demographic transition is a theory, a generalization, a framework for analysis or merely an 'idea'. Or is it an 'historical model, predictive model, or a mere descriptivte term'? The debate about thes status of transition theory continues to occupy a central place in demography.
In any case, there's no claim here (of Gates) precise enough to investigate further.