Could Ned have managed to form an alliance with the Martells against the Lannisters after Robert’s death, had he held on just a little while longer?

No, but it’s not his fault, it’s a question of different motives. 

At the end of the day, while Ned wants to protect his nephew’s life, he doesn’t want Jon Snow on the Iron Throne. If he did, he wouldn’t have let him go to the Wall and take his vows with the Night’s Watch, disinheriting himself. 

When it comes to a decision, Ned is firm that Stannis is the rightful king, as Robert Baratheon’s trueborn heir. By contrast, Doran Martell is a secret Targaryen loyalist who wants a Targaryen on the Iron Throne. He would not, indeed did not, support Stannis. To do that is to recognize that the Usurper, the man who (as Doran sees it) legitimized the murder of his sister, niece, and nephew was rightfully King of Westeros.