Does assigning missing literals after unit propagation keep a Boolean formula satisfiable? Let's denote $F$ a Boolean formula (in CNF if you wish) and $F_{a}$ the same formula where literal $a$ was assigned and propagated (using unit propagation).

If $F_{a} \rightarrow b$ (b is propagated from a) and $F_{\neg a} \rightarrow b$ then I can imply $b$ ($\neg a \rightarrow b \land a \rightarrow b \Rightarrow b $). In other words, any satisfying assignment for $F$ must have $b$.

If $F_{a} \rightarrow b$ and $F_{\neg a}$ is independent of $b$, is there always a satisfying assignment for $F$ that has $b$ when $F$ can be satisfied?
If "independent" means "not inferred by unit propagation", then the answer is "no". Eg. 
$$ 
\lnot a \lor b, x\lor \lnot b,\lnot x\lor \lnot b
$$.