I don't think FW is interested romantically in Louisa at all at this point, but is doing what he feels he should do, since he feels responsible for the accident. I think any doubts he'd had had been doubly reinforced by her stubborn behavior leading up to her "fall." (I call it a jump).

And if he were to sit vigil and not move from the house during her first week of illness (when she was only awake for brief moments, in which, as has been pointed out, he would not have been admitted to see her anyway), what good would it have done Louisa? She is not awake to appreciate his efforts and is so out of it, I'm sure she cares not whether he is within the Harville's roof or not. After he visits Kellynch briefly he leaves for Lyme "resolving not to quit it again." The point is that she has gotten a bit better, and I imagine that he feels that in the near future his presence at Louisa's side might actually be allowed and appreciated.

He's shown the most concern he possibly can for Louisa. He did what he could to help her family, by taking Henrietta back to Uppercross and taking the news to her parents, a job which he feels bound to do (and something, as a ship's captain, he's probably quite used to doing). This is what a man was expected to do for the sick. I should think his navy peers would rather commend him on his thoughtful efforts, which really should have fallen to Charles, but he was compelled to stay since I suppose it was imagined that she might suddenly die and have no one there from her family.