Societies aren't static, even when everyone is immortal.
There's also no guarantee that we have further moral progress in the future, especially when you consider the last century's two global wars.
The last 10 years has seen major global progress, let alone the last century, wars included.
People always love to think they're at the pinnacle. The future will be far more fantastic than either of us could possibly imagine, because it will be imagined by people who grow up learning from the mistakes we make today.
Also, last week's UK Remain vote is a pretty good indicator how old and young people think different. Young people today consider themselves more global, while old people are afraid to loose their national identity and pride. Both are OK, but as long as the young keep replacing the old the current path of humanity will, in a few generations perhaps, hopefully lead to global unification. On the other hand, if everyone suddenly became immortal it would put the world in stasis as the ones who currently hold the wealth (overwhelmingly the boomer generation) would cling to what they own and their ways of thinking instead of passing on the torch so to speak.
If humanity discovers immortality before we become an expanding multi-planet and post-scarcity society it will be an absolute disaster. Because immortality doesn't change human nature.
Imagine a boot stamping on a human face - forever.