"the geometry near massive objects is not Euclidean"
Ok, I'll bite.
The bending of space/time means that space is "non-Euclidian" everywhere, but the effects are more noticeable closer to massive objects, and as close-to-negligible-as-makes-no-difference away from them.
There's no contradiction. The same effect is happening everywhere. It's just hard to observe or measure in most places.