Birth rates tend to stabilize in first world countries anyway. The only reason that isn't happening in the US is immigration--legal and otherwise. In Japan, the population is decreasing. In Europe, it's near zero growth in many countries.
Some people do have a lot of kids. However, the overall total fertility rate in the US was estimated in 2011 at 1.89 kids/woman, below replacement level. So without immigration, population would indeed be shrinking.
It sounds like you're talking about 'immigration raising the population', which is slightly orthogonal to the grandparent post statement that birth rates tend to stabilize, but are higher than otherwise because of immigration. I assume what he was actually referring to was the tendency of first generation immigrants to have a higher birth rate than non-immigrants in America.
