Can Tarjan's SCC algorithm find satisfying assignment to just 'any' digraph? Is Tarjan's algorithm capable of finding satisfying assignment to any digraph consists of variables (vertices) and implications (edges)? I know that it solves implication graphs constructed by 2SAT clauses, but I wonder if the algorithm can handle the digraphs which cannot be expressed as 2SAT clauses. Let me clarify this by the following examples:

(a+b)(~b+c) translates to the following
(~a -> b),
(~b -> a),
(b -> c),
(~c -> ~b)
It is OK. But how about the following implications:
(a -> b),
(b -> ~c)
There is obviously no corresponding set of 2SAT instances. Does Tarjan's algorithm find a satisfying assignment to this?
You have a faulty premise.  There is a 2SAT instance corresponding to $(a \implies b) \land (b \implies \neg c)$.  Just note that $x \implies y$ is equivalent to $\neg x \lor y$.  Then you can transform the original formula to $(\neg  a \lor b) \land (\neg b \lor \neg c)$, which is obviously a 2SAT instance.

That algorithm you refer to can solve 2SAT, but it can't solve anything stronger as is.  However, there are related graph-based algorithms to test satisfiability of arbitrary conjunctions of Horn clauses.