Here's an algorithm that is almost A* search. Essentially it's BFS with a priority queue that uses A* priority.
frontier <- empty priority queue
frontier.insert(start) with priority g(start)+h(start)
while frontier isn't empty
vertex <- dequeue frontier
for each undiscovered neighbor of vertex
neighbor.discovered = true
neighbor.parent = vertex
frontier.insert(neighbor) with priority g(neighbor)+h(neighbor)
if neighbor == goal, stop
This algorithm is missing the parts of A* that handle this fact: the first path found to a vertex is not necessarily the shortest path to that vertex.
It's easy to come up with examples where the missing parts are crucial... for weighted graphs. For unweighted graphs, I haven't been able to come up with any.
Is it possible that this simpler version of A* is correct for unweighted graphs?
No, it is not correct for an arbitrary h function. Here is an example. Let's assume that we have a graph with 7 vertices and the following unweighted edges: {(1, 2), (2, 3), (3, 4), (4, 6), (2, 5), (5, 6), (6, 7)}. We can define h in the following way: {0, 0, 0, 0, 2, 1, 0}. The start vertex is 1, the goal is 7. It is easy to verify that this heuristic function is admissible. However, if we run this algorithm we will see that the path to the 6 vertex that it finds first is (1, 2, 3, 4, 6) because f(4) = 3 + 0 < 2 + 2 = f(5), while the actual shortest path to it is (1, 2, 5, 6).