Did you post a picture? I don't see one. Try posting one here in the comments section.

You could hide it with a skirt or fringe

you already have a black chair against ivory, it's part of the chair.

OP said the chair is ivory?

Post pics please

Rustoleum spray paint in any color you want?

It's the nature of the beast. If you are willing to give up the rocker, there is something like this one at Bed Bath and Beyond.

Your house and your guests, but as a guest in my friend's house, who has the rocker/swivel chairs, I hated to sit in them. Made me feel odd; add to that all she did was swivel, rock, swivel, rock and I got sea sick just watching her.

Is this an item you just bought and can return? Your didn't show your chiar particular model but there are several swivel models on which the base doesn't show at all, or it's a wood base, some have a light colour base similar to your, etc.

Spray paint the base whatever color looks better if you don't want a skirt and hate the black base.

Add more black accents in the room so it does not stand out...blends in. Table legs, picture frames, decor, lamp bases, etc