If they were immortal they'd soon find a way to move beyond this planet, then beyond this solar system. Once that is accomplished there is a whole universe to colonise so I assume that is what would happen. It would be the only way really since the planet would otherwise quickly fill up and 'immortality' would end up a pipe dream due to endless conflict over land and sustenance.
That's far from guaranteed.
With an average fertility rate of 1, which is higher than that of many countries including where I live, even complete immortality would lead to an asymptotic population. Since some people will still inevitably die in accidents even if aging is solved, a fertility rate of 1 would be a declining population even with unlimited lifespans.
You're assuming everyone gets the immortality and is de facto benevolent. If you actually look at how the super rich and powerful have acted historically - its much more likely that the few richest/most powerful people would just take control of the earth, reserve immortality for themselves, and keep everyone else as a mortal slave.
Ah yes, the "vampire spaceship" season of humanity.