To answer your question directly, no, .NET Core isn't dependent on WINE and it wouldn't really be a compliment or enhance it.
WINE is the Win32 APIs implementation for Linux and allows you to run native Windows programs on Linux whereas .NET Core allows you to run managed programs on Linux, Mac and Windows and in the future maybe more, just to emphasize it doesn't uses WINE at all, so your assumption is correct.
However, you can think about this as a compliment in the sense that you get two platforms that run Windows applications on Linux.
p.s. I think that with Mono you can actually do P/Invoke to Windows APIs through WINE but I never tried it so dunno.