Is it true that white people (Caucasians with no Slavic ancestry) are descended from "barbarians"?

The Romans considered everyone who was not a Roman as a 'barbarian.' that would include the bulk of europe until the Romans took over that country and Romanised it. You are talking about a lot of different peoples here from the Mediterranean to Scandinavia.Most Europeans descend from either hunter gatherers or neolithic farmers long before Roman times, and the population movement tended to come across eastern and central Europe and up via the mediterranean. Scandinavia has some old genes, but remember it would have been glaciated a long time in the Ice age and populations were sheltering in places like Spain,from which they spread afterwards.