I would also appreciate if it was as far from the physicists formalism as possible, no abstract indices ,etc. Also I don't consider using a basis or tetrads as coordinate free. The idea is to use only a clean abstract purely geometrical language without encoding operations with indices or matrices of coordinates.
Try "General Relativity for mathematicians" by R. Sachs and H. Wu. Also, "Gravitation" by C.W. Misner, K.S. Thorne, J.A. Wheeler - it's so famous that it's got its own Wikipedia page. Finally, "The large scale structure of space-time" by S.W. Hawking and S.F.R. Ellis - another "star" with a Wikipedia page. All of them were published in the '70s, so they might not be up to date with the experimental part (thet you are probably not interested in, given your question) or with the back-hole cosmology. But once you're proficient in the subject, you'll be able to find your way further by yourself.
Also, notice that a coordinate-free approach is an extremist dream, that I warmly invite you to get rid of as soon as possible (I've been there too, but now I'm cured). If you're unhappy with the above books, I'm afraid that they are as coordinate-free as it gets. Even Riemannian geometry is often done with a mature mix of coordinate and invariant methods. Good luck defining volume forms without coordinates!
After a lot of searching I came across Advanced general relativity lecture notes by Sergei Winitzki and an intro to differential geometry and curvature by Hestenes ,link below,. Misner's Gravitation is likely the best relativity book but it's only partly coordinate and index free.