There is one event that could definitively prove that Jon is a legitimate Targaryen heir, and that event is the on-screen wedding of Rhaegar and Lyanna.
If they were, Jon is a legitimate Targaryen and ahead of his aunt Dany in the line of succession.
Assuming Rhaegar + Lyanna = Jon, the big question is whether or not Rhaegar and Lyanna were married after they ran off together.