Yes I guess some believe that, but it's pretty dumb.
It's in no way obvious that abortion causes a lower birth rate, as it may hinder the ability of women to have subsequent births. It is also the case that it's not immediately obvious which ethnic groups are most affected by abortion rules, though people who consider the issue likely have a good idea.
My current mental model is that pro-lifery is the equivalent of political "junk DNA". It made sense for GOP politicians in the 1980s, 19990s, and 2000s because it was a genuine vote getter. I think at the time more people were really concerned about saving babies, God implanting the soul into the body at conception, etc. Now, people often think differently, but the politicians have a legacy of ideology to uphold.
Myself, I am pro-life in my own way. I would prefer policies that tackle the opioid, suicide, firearms, and traffic epidemics killing far in excess of 100,000 Americans prematurely a year. I'd also like a cleaner food, water, and ventillation program that would probably save even more lives.