The key is to have a large primary culture pop and enact health care reforms early on to grow it as much as possible. Once the colonies become states, way more primary pops start appearing there and they color your culture map. I had a full-citizenship party in power for a good chunk of the game, but I'm not sure that made a difference.
It's also worth mentioning that it's nigh impossible to do the same if you're playing PDM. The colonial administration decisions/ events stop it happening by assigning a nation with the primary culture of the natives, which prevents them assimilating at all.