Assuming that we don't crack the immortality problem, a population can continue to have children without that population growing. A population of 8 billion for example with sufficiently good medical care that the average person lives to 75 could have about 100 million children a year without having net population growth.
If a population is depending on continued net growth it has to very careful. If there is some constant above which it needs to maintain its growth rate growth will be an exponential process and exponential processes can blow up on you way faster than you might expect.
As an example, suppose we were depending on 1% annual population growth indefinitely and bad things would happen if we could not sustain that. Let's try to put an upper limit on how long we could theoretically sustain that.
Assumptions: we do not develop faster than light travel, do not discover any extra dimensions of spacetime that we can occupy, every human now in existence is near Earth, we don't need space for anything other than holding people, and you can't pack people tighter than 16 people per m^3 (the number of present day people you could fit in a 1m x 1m x 1m cube if we were versatile shape shifters who could take on arbitrary shapes but not change our volume).
Given those assumptions, then N years from now every human in existence must be within N lightyears of Earth. Take the volume of that sphere at year Y and divide it by the number of people that would be alive in year Y at 1% annual growth and we get how much space is in the most optimistic case available per person.
Q: How long before that space per person is less than 1/16th of a cubic meter?
A: ~12000 years.
If we can drop the growth rate to 0.5% we get about 24500 years. If we can drop it to 0.1% we get around 127000 years.
Exponentials are scary.