I don't think Stannis is doing what he is doing out of duty or inherit greed, but because of his ego.
"Whatever he did, Robert had done first, and better."
This explains his motivation. Robert was a better warrior, more handsome, more charismatic, and arguably even a better commander.
However, Robert was a shit king. In this regard, Robert would be king first, but Stannis would be king better.
"All the same, the king had been on the point of refusing them until Lord Bryce Caron said, "Your Grace, if the sorceress is with us, afterward men will say it was her victory, not yours. They will say you owe the crown to her spells." That had turned the tide. Davos himself had held his tongue during the arguments, but truth be told, he had not been sad to see the back of her."
This strikes me as odd. Why does Stannis care so much if people think he won because of a sorceress or not?
Its because Robert won his war with his warhammer alone. He didn't need dragons, or magic or any of that crap. Stannis being better than Robert in a regard would be absolutely ruined if they found out he "cheated" in a sense.
And we know that Stannis has ISSUES with Robert. His entire crest for the crown seems to be one gigantic ego trip.