Simple answer: Because Jon would still be in no position to claim the throne, and no one would believe him anyway.
Longer answer: Jon is, what, 14 in the books? While I think that's old enough in the books to rule in his own name (might be 16 -- I'm not sure), it's also possible that he'd end up in a regency much like Tommen. Moreover, Jon is still vulnerable. He doesn't hold claim to the North (and wouldn't anyway, as the son of a Stark daughter, I think), and there are virtually no Targaeryan loyalists left. Well, no open Targaeryan loyalists. There are the Martells, but it remains to be seen how well that'll hold up. It's possible the Tyrells might, but they're iffy. Plus, unless Rhaegar and Lyanna were secretly wed (which IS a possibility), he's still a *******, just a ******* Starkgaeryan rather than a ******* Starkwhorian. In other words, his claim is weak legally, and he himself is weak militarily, especially at the end of the first book. And he's all the way in the North at the Wall. And Ned (and Howland Reed) are the only survivors of the Tower of Joy that we know of, and both would be questionable sources, even though we, the readers/viewers trust Ned.
If anything, I think Dany marrying Jon helps to secure far more support for her. Theoretically, at least within the world of the show where Doran is dead, Ellaria Sand is alive, Yarra wants to become queen, the Tyrells have PTFO of King's Landing, the North is in disarray but Sansa has taken control, the Vale and (ostensibly) the Riverlands are under Petyr's control and he's (currently) backing Sansa, and Sansa is positioned to marry Robin and rule the North and de facto the Vale, Jon is a MUCH better prospect than he is in the books where they stand currently. Dany could, in theory, gain the support of the majority of the Seven Kingdoms with two alliances and one marriage, and maybe six out of seven if she can convince the Tyrells. Of course, the Lannisters will fight on at least until Cersei and Tommen die. Jaime, if he survives, would, I think, be content to rule the Westerlands and marry, although that'll require....well...a few other things to happen.