It sure was.

Yes, it was explicitly part of American culture, and is written into our very foundational documents.

If you include the period before we split from England we spent centuries taking land that didn't belong to us and building our economy at least partly on slavery. If you look at the South in that time frame slavery permeated their culture. It is as fair to say that slavery was if anything much more deeply embedded in Southern culture than whaling in Iceland.

Except can you really say “genociding native americans”

As a country, the US has spent more of its existence genociding native Americans than allowing women to vote, or having a standing army.

and “slavery” are a part of American culture?

The US currently has fully legalized privatized slavery. You, specifically you, can own a slave in the US right now. You can even treat them as if the constitution does not apply to them in any way. Simply buy a prisoner and get a judge to commit that prisoner to you for the length of their sentence. It's so ingrained in our culture, we've never stopped the practice.