Copying from the interview, here's Harry's actual words about that conversation. The closest Harry gets to anything race-related is to say he was asked 'what will the kids look like' when Oprah put that sentence to him.
Oprah: Well, what is particularly striking is what Meghan shared with us earlier, is that no one wants to admit that there’s anything about race or that race has played a role in the trolling and the vitriol, and yet Meghan shared with us that there was a conversation with you about Archie’s skin tone.
Harry: Mm-hmm.
Oprah: What was that conversation?
Harry: That conversation I’m never going to share, but at the time . . . at the time, it was awkward. I was a bit shocked.
Oprah: Can you . . . can you tell us what the question was?
Harry: No. I don’t . . . I’m not comfortable with sharing that.
Oprah: OK.
Harry: But that was . . . that was right at the beginning, right?
Oprah: Like, what will the baby look like?
Harry: Yeah, what will the kids look like?
Oprah: What will the kids look like?
Harry: But that was right at the beginning, when she wasn’t going to get security, when members of my family were suggesting that she carries on acting, because there was not enough money to pay for her, and all this sort of stuff. Like, there was some real obvious signs before we even got married that this was going to be really hard.