Can I mix entirely on headphones?

You can get 90% there, and plenty of released music proves it. Headphones exaggerate stereo width and hide how bass sums in a room, so the discipline is: mix on open-backs, reference on known commercial tracks constantly, and check the low end and mono compatibility on any speaker — even a car. Room-simulation plugins narrow the gap further. What you cannot do is skip the referencing and trust the first impression.