We have many examples whereby societies that become "comfortable" see decreasing birth rates leading to negative population growth. As mortality pressure decreases, the urgency and drive to hurry up and have children decreases.
There's no reason to believe that if we all suddenly could live forever that we wouldn't dramatically slow down how quickly we had children.
Talk to any family about why they're stopping to have children. You'll normally get answers along the lines of, "Well, we're not getting any younger." What if you know that you're not getting any older either?
Personally, if my wife and I hadn't been pushing into our 30's and worried about having a couple of kids before we were too old, we would have waited indefinitely.
Who knows... maybe we might not even have been married? If you have 1,000's of years ahead of you, do you saddle yourself with one person for an eternity?
Population growth has been declining since the 60s.
UN estimates are that we'll peak at 10 billion.
Besides that it's already happening, reducing population growth isn't necessarily a good thing, in econimic terms of aggregate "utility." Is there a difference between preventing birth and causing death?
That estimated peak is based on existing and projected death rates. Even Japan's population would grow if death from old age and age-related disease ceased.
Of course that means we need an ethical, voluntary, way of keeping population in check.
Which infers true sexual equality, or you get one child policies as successful as China's.
Or we have an exponential increase in lifespan and ever more people chasing those finite resources.
So, it's going to be a world very different from ours, or it's going to be a bloody mess.
May you live in interesting times?