Can I drink a bottle of white wine after ingesting a litre of red wine, or will it make me really sick?!

Yes --- drinking 2 bottles of any wine (within a few hours or so) will either make you sick, or very very sleepy and maybe both.

It sounds risky to me, given how many people have issues with either red or white wine. Also, are you used to drinking 2 bottles of wine in a night? That seems like a lot.
Still, I know of no hard and fast rule about mixing wines...otherwise there would probably never be wine pairing parties where they try all sorts of wines in one evening.

You can mix wines with no problem, I think you'd get sick for downing that much wine regardless of the color.

That will make you more sick than drinking 2 bottles of the same kind of wine.

The amount will make you sick, not the mixing.