Also you have the problem of where you put this growing population of immortals? It seems that you would have to solve the problem of interplanetary colonization before we'd be ready to solve death.
These things may actually be tangled together. Interplanetary travel is likely to cause some radiation damage to the bodies of the astronauts. Radiation damage manifests a lot like premature aging; these two may have something in common.
In that case, colonizing planets and trying to treat aging will have to develop together, much like development of airplanes and weather forecasting did.
Unless you can prevent people from traveling overseas to receive anti-aging treatments, clinics will spring up from Costa Rica to Indonesia. The US, even if it wanted to protect the IP of the few very hard, will have to weigh it against the risk that a systemic competitor like China will provide treatments for artificially low price just out of spite and to gain some sympathy points.
Look at the Cold War. During the conflict with the Communist Bloc, inequality in the West actually went down. In presence of ideological competitors, Western governments learnt to mitigate the worst excesses of their own systems.
But in all seriousness, I agree that being able to prevent aging and mortality likely would not lead to paradise. Probably access to this technology would not be distributed equitably, and we would see an elite few living forever, with mere mortals resigned to a disposable under-class.